Rigorous Differentiation and Integration

The Central Idea

Differentiation studies local linear behavior. Integration studies accumulation through increasingly fine approximations. Both are limiting processes, so their rigorous development depends on the language of convergence, continuity, compactness, and completeness.

The central relationship is

From Computational Calculus to Analysis

In a computational course, we often begin with formulas such as

or

Analysis begins one level deeper. We ask:

• What does the derivative actually mean?
• When does a derivative exist?
• Which functions are Riemann integrable?
• Why do partitions approximate area correctly?
• When may limits, derivatives, and integrals be interchanged?
• What hypotheses make the Fundamental Theorem valid?

The Derivative at a Point

Let \(f\) be defined on an interval containing \(a\). The derivative of \(f\) at \(a\) is

provided this limit exists as a finite real number.

The quotient

is a difference quotient. The derivative is the limiting slope of these secant lines as the second point approaches \(a\).

An Equivalent Difference-Quotient Form

Writing \(x=a+h\) gives the equivalent definition

Both definitions express the same limiting process. One emphasizes the increment \(h\); the other emphasizes a nearby point \(x\).

Example: Deriving the Derivative of \(x^2\)

Let

At \(a\),

Expand:

Therefore,

Differentiability Is Stronger Than Continuity

If \(f\) is differentiable at \(a\), then \(f\) is continuous at \(a\).

To see the structure, write

As \(x\to a\), differentiability gives

while

Hence

so \(f(x)\to f(a)\).

The Converse Is False

Continuity does not imply differentiability.

The function

is continuous at \(0\), but

while

Since the one-sided limits disagree,

One-Sided Derivatives

At an endpoint, or when investigating a corner, one-sided derivatives are useful:

and

At an interior point, the ordinary derivative exists exactly when both one-sided derivatives exist and are equal.

Differentiability as Linear Approximation

The derivative is more than a slope. It gives the best first-order linear approximation to a function near a point.

If \(f\) is differentiable at \(a\), then

where the remainder satisfies

Equivalently,

The Meaning of \(o(h)\)

The notation

means

Thus the error in the linear approximation is small compared with the size of the input increment itself.

Derivative Rules as Theorems

Once differentiability is defined by limits, familiar rules follow from limit laws.

If \(f\) and \(g\) are differentiable at \(a\), then

and, when \(g(a)\ne0\),

Why the Product Rule Works

Begin with

Add and subtract \(f(a+h)g(a)\):

Rearranging gives

Differentiability implies continuity, so \(f(a+h)\to f(a)\). Taking limits yields the product rule.

The Chain Rule

Suppose \(g\) is differentiable at \(a\) and \(f\) is differentiable at \(g(a)\). Then

The rigorous proof requires care because the usual quotient manipulation appears to divide by \(g(a+h)-g(a)\), which may occasionally be zero. A proof based on the linear-approximation form of differentiability avoids that difficulty cleanly.

Fermat's Theorem on Interior Extrema

Suppose \(f\) has a local maximum or local minimum at an interior point \(c\), and suppose \(f'(c)\) exists. Then

This theorem explains why differentiable interior extrema must occur at critical points. It does not say that every point with derivative zero is an extremum.

Rolle's Theorem

Suppose \(f\) is continuous on \([a,b]\), differentiable on \((a,b)\), and

Then there exists \(c\in(a,b)\) such that

The compactness of \([a,b]\) helps ensure that a continuous function attains maxima and minima, while differentiability converts an interior extremum into a zero derivative.

The Mean Value Theorem

Suppose \(f\) is continuous on \([a,b]\) and differentiable on \((a,b)\). Then there exists \(c\in(a,b)\) such that

The instantaneous rate of change equals the average rate of change at least once somewhere inside the interval.

Why the Hypotheses Matter

The Mean Value Theorem requires:

• continuity on the closed interval \([a,b]\);
• differentiability throughout the open interval \((a,b)\).

A jump discontinuity, a corner, or another failure of differentiability may destroy the conclusion. In analysis, hypotheses are part of the theorem, not decorative assumptions.

Consequences of the Mean Value Theorem

If

throughout an interval, then \(f\) is constant on that interval.

If

throughout an interval, then \(f\) is strictly increasing there.

If

then for all \(x,y\) in the interval,

Thus a bounded derivative produces a Lipschitz bound.

Cauchy's Mean Value Theorem

Suppose \(f\) and \(g\) are continuous on \([a,b]\) and differentiable on \((a,b)\). Then there exists \(c\in(a,b)\) such that

When \(g'(c)\ne0\) and \(g(b)\ne g(a)\), this may be written

This theorem is one route to rigorous versions of l'Hopital-type results.

Darboux's Theorem for Derivatives

Derivatives need not be continuous. Nevertheless, every derivative has the intermediate value property.

If \(f'\) takes values \(u\) and \(v\) at two points, then every value between \(u\) and \(v\) is also attained by \(f'\) somewhere between those points.

Therefore a derivative cannot have a jump discontinuity.

Higher Derivatives

If \(f'\) is differentiable, its derivative is the second derivative:

Continuing gives

Higher derivatives encode increasingly refined local behavior, but their existence must be established rather than assumed.

Taylor's Theorem

If \(f\) has enough derivatives near \(a\), then \(f\) can be approximated by a polynomial built from derivative data:

Under standard hypotheses, one form of the remainder is

for some \(\xi\) between \(a\) and \(x\).

Rigorous use of a Taylor polynomial therefore requires control of the remainder, not merely calculation of the polynomial.

Turning from Differentiation to Integration

Differentiation begins with local quotients. Riemann integration begins with finite sums over partitions and asks whether those sums stabilize as the partition becomes increasingly fine.

The key idea is approximation by rectangles, but the rigorous definition must guarantee that the final value does not depend on arbitrary choices of sample points.

Partitions of an Interval

A partition \(P\) of \([a,b]\) is a finite ordered set

satisfying

The Mesh of a Partition

The mesh, or norm, of a partition is

Saying

means that even the largest subinterval is becoming small.

Riemann Sums

Choose a sample point

from each subinterval. The corresponding Riemann sum is

A Riemann integral exists when these sums approach one common value as the partition becomes fine, regardless of the allowed sample-point choices.

Riemann Integrability

A bounded function \(f:[a,b]\to\mathbb{R}\) is Riemann integrable if there exists \(I\in\mathbb{R}\) such that for every \(\varepsilon>0\), there is \(\delta>0\) with the property that

whenever the tagged partition satisfies

In that case,

Upper and Lower Darboux Sums

For a bounded function \(f\) and partition \(P\), define

The upper sum is

and the lower sum is

The Darboux Integrability Criterion

A bounded function \(f:[a,b]\to\mathbb{R}\) is Riemann integrable if and only if for every \(\varepsilon>0\), there exists a partition \(P\) such that

The condition says that the total uncertainty between upper and lower rectangle approximations can be made arbitrarily small.

Refinement Improves Bounds

If \(Q\) is a refinement of \(P\), then

Adding partition points can only raise lower sums and lower upper sums. The two families of approximations squeeze the possible integral value.

Continuous Functions Are Riemann Integrable

Every continuous function on a closed bounded interval \([a,b]\) is Riemann integrable.

The logic is structural:

1. \([a,b]\) is compact.
2. A continuous function on a compact set is uniformly continuous.
3. Uniform continuity makes the oscillation of \(f\) small on sufficiently short subintervals.
4. Therefore upper and lower sums can be made arbitrarily close.

Monotone Functions Are Riemann Integrable

Every bounded monotone function on \([a,b]\) is Riemann integrable, even if it has discontinuities.

Thus continuity is sufficient for Riemann integrability, but it is not necessary.

Finite Discontinuities Do Not Prevent Integrability

A bounded function with only finitely many discontinuities on \([a,b]\) is Riemann integrable.

The discontinuous points can be enclosed in intervals of very small total length, so their contribution to the difference between upper and lower sums can be controlled.

A Bounded Function That Is Not Riemann Integrable

Define the Dirichlet function

Every nonempty interval contains both rational and irrational numbers. Therefore, on every subinterval,

Hence

and

for every partition \(P\). The upper and lower sums never approach one another, so \(f\) is not Riemann integrable.

Basic Properties of the Riemann Integral

If \(f\) and \(g\) are Riemann integrable on \([a,b]\), then

and for \(a \[ \int_a^b f = \int_a^c f+\int_c^b f. \]

Order and Comparison

If

for all \(x\in[a,b]\), then

In particular, if \(f\ge0\), then

An Integral Absolute-Value Bound

For an integrable function,

More generally, if \(|f(x)|\le M\), then

The Mean Value Theorem for Integrals

If \(f\) is continuous on \([a,b]\), then there exists \(c\in[a,b]\) such that

Therefore the average value

is actually attained by \(f\) somewhere on the interval.

Accumulation Functions

Given an integrable function \(f\), define

This function records the accumulated signed contribution of \(f\) from the fixed starting point \(a\) to the variable endpoint \(x\).

Continuity of the Accumulation Function

If \(f\) is Riemann integrable on \([a,b]\), then the accumulation function

is continuous.

If \(|f(t)|\le M\), then

Thus \(F\) is actually Lipschitz.

Fundamental Theorem of Calculus, Part I

Suppose \(f\) is continuous on \([a,b]\), and define

Then \(F\) is differentiable on \((a,b)\), and

Integration followed by differentiation recovers the original continuous function.

Why Part I Works

Consider the difference quotient:

By the Mean Value Theorem for integrals, there is a point \(c_h\) between \(x\) and \(x+h\) such that

As \(h\to0\), we have \(c_h\to x\). Continuity gives

so \(F'(x)=f(x)\).

Fundamental Theorem of Calculus, Part II

Suppose \(f\) is continuous on \([a,b]\), and let \(F\) be any antiderivative of \(f\):

Then

This theorem converts an accumulation problem into endpoint evaluation of an antiderivative.

The Newton-Leibniz Connection

The Fundamental Theorem explains why differentiation and integration behave as inverse processes. The relationship is not merely computational; it is a theorem linking two independently defined limiting constructions.

for continuous \(f\), while

Integration by Parts

The product rule implies

Integrating and applying the Fundamental Theorem gives

Thus integration by parts is the integrated form of the product rule.

Substitution as a Theorem

A rigorous substitution rule follows from the chain rule and the Fundamental Theorem. Under appropriate differentiability and continuity hypotheses,

The validity of a substitution therefore depends on the regularity of the functions involved, not merely on a symbolic replacement of variables.

Uniform Convergence and Integration

Suppose \(f_n\to f\) uniformly on \([a,b]\), and each \(f_n\) is Riemann integrable. Then \(f\) is Riemann integrable and

The proof uses

and uniform convergence drives the right side to zero.

Uniform Convergence and Differentiation

Differentiation is more delicate than integration.

Uniform convergence of \(f_n\) alone does not imply that

A useful theorem is the following: if each \(f_n\) is differentiable, \(f_n(x_0)\) converges at one point \(x_0\), and \(f_n'\) converges uniformly to a function \(g\), then \(f_n\) converges uniformly to some differentiable function \(f\) and

A Warning Example for Differentiation

Consider

Since

we have

uniformly. But

which does not converge pointwise for many \(x\), much less uniformly. Uniform convergence of functions therefore does not control derivatives.

Differentiation Under the Integral Sign

Suppose a function depends on both a parameter \(x\) and an integration variable \(t\):

Under suitable regularity and uniformity conditions, one may differentiate through the integral:

Later analysis courses develop broader versions of this principle using measure-theoretic convergence theorems.

Improper Integrals Are Different

A Riemann integral on a bounded closed interval is defined directly from partitions. An improper integral is instead defined by a limit of proper integrals.

For example,

means

when that limit exists.

The existence of each finite-interval integral does not by itself guarantee convergence of the improper integral.

Riemann Integration Has Limits

The Riemann integral handles continuous functions and many discontinuous functions very well, but it is not flexible enough for every limiting process used in modern analysis.

Important questions remain:

• How should we integrate highly discontinuous functions?
• When can pointwise limits of integrable functions remain integrable?
• How should sets of negligible size be treated?
• How do we build function spaces suited to convergence in average?

These questions motivate measure theory and the Lebesgue integral.

Common Error: Assuming Continuity Implies Differentiability

The implication

is valid, but the reverse implication is not.

The standard counterexample is

at \(x=0\).

Common Error: Ignoring Theorem Hypotheses

Before applying Rolle's Theorem or the Mean Value Theorem, check both interval conditions:

• continuity on the closed interval;
• differentiability on the open interval.

A correct formula cannot compensate for a missing hypothesis.

Common Error: Treating Every Bounded Function as Integrable

Boundedness is necessary for the standard Riemann definition on \([a,b]\), but boundedness alone is not sufficient.

The Dirichlet function is bounded between \(0\) and \(1\), yet it is not Riemann integrable.

Common Error: Using One Riemann Sum as the Definition

A single left-endpoint, right-endpoint, or midpoint sum is only an approximation.

The integral is the limiting value that survives arbitrary sufficiently fine tagged partitions.

Common Error: Interchanging Limits Without Justification

Expressions such as

and

are not automatically equal.

Likewise,

need not equal

Interchanging analytic operations requires a theorem whose hypotheses have been verified.

A Reliable Procedure for Proving Differentiability

1. Identify the point \(a\) and write the difference quotient.
2. Simplify algebraically without substituting \(h=0\).
3. Determine whether the limit exists and is finite.
4. Check one-sided behavior when a corner, endpoint, or piecewise definition is involved.
5. State the derivative only after the limit has been established.
6. Use differentiability to infer continuity when useful.

A Reliable Procedure for Applying a Mean-Value Theorem

1. Identify the closed interval \([a,b]\).
2. Verify continuity on \([a,b]\).
3. Verify differentiability on \((a,b)\).
4. Compute the relevant endpoint quantity.
5. State the theorem's conclusion precisely.
6. Solve for or characterize the guaranteed point \(c\) when possible.

A Reliable Procedure for Proving Riemann Integrability

1. Verify that the function is bounded on \([a,b]\).
2. Look for a standard sufficient condition such as continuity or monotonicity.
3. If needed, use upper and lower Darboux sums.
4. Control the oscillation on each subinterval.
5. Construct a partition making \(U(f,P)-L(f,P)\) smaller than a prescribed \(\varepsilon\).
6. Conclude integrability from the Darboux criterion.

A Reliable Procedure for Interchanging a Limit and Integral

1. Identify the sequence or family of functions.
2. Determine the type of convergence available.
3. Check whether the relevant theorem requires uniform convergence or another stronger condition.
4. Verify integrability of the approximating functions.
5. Apply the theorem only after its hypotheses are satisfied.
6. State the exchanged limit explicitly.

Example: Mean Value Theorem Workflow

Let

on \([1,4]\). Since polynomials are continuous and differentiable, the Mean Value Theorem applies.

The average slope is

Since

we solve

to obtain

Example: Darboux Criterion for \(f(x)=x\)

Consider \(f(x)=x\) on \([0,1]\) and divide the interval into \(n\) equal subintervals of width \(1/n\).

Because \(f\) is increasing, the oscillation on each subinterval is

Therefore

Since

the Darboux criterion confirms that \(f\) is Riemann integrable.

Example: Fundamental Theorem Workflow

Define

The integrand is continuous, so the Fundamental Theorem gives

An antiderivative is

so

Practice 1: Derivative from the Definition

Use the limit definition to find the derivative of

at an arbitrary point \(a\).

Practice 2: Continuity versus Differentiability

Determine whether

is continuous and differentiable at \(x=2\).

Practice 3: Rolle's Theorem

Verify the hypotheses of Rolle's Theorem for

on \([1,3]\), and find a point \(c\) guaranteed by the theorem.

Practice 4: Mean Value Theorem

Apply the Mean Value Theorem to

on \([0,2]\). Find all values \(c\in(0,2)\) satisfying the conclusion.

Practice 5: Lipschitz Bound from a Derivative

Suppose \(f\) is differentiable on an interval and

throughout that interval. Prove that

Practice 6: Darboux Integrability

Explain why every continuous function on \([2,5]\) is Riemann integrable. Your explanation should identify the role of compactness.

Practice 7: A Nonintegrable Function

Let

on \([0,1]\). Determine its upper and lower Darboux sums and decide whether it is Riemann integrable.

Practice 8: Fundamental Theorem of Calculus

If

find \(F'(x)\).

Practice 9: Uniform Convergence and Integration

Suppose \(f_n\to f\) uniformly on \([0,3]\). Show that

Explain why the integral difference tends to zero.

Practice 10: Identify the Error

A student argues:

Since \(f_n\to f\) uniformly and every \(f_n\) is differentiable, \(f_n'\to f'\).

Explain why the conclusion does not follow from the stated hypotheses.

Practice Answers

Practice 1

Begin with

We have

Therefore,

For \(h\ne0\),

Taking \(h\to0\) gives

Practice 2

The absolute-value function is continuous, so \(f\) is continuous at \(x=2\). For the derivative,

The right-hand limit is \(1\), while the left-hand limit is \(-1\). Therefore,

Practice 3

The polynomial is continuous on \([1,3]\) and differentiable on \((1,3)\). Also,

Since

Rolle's Theorem guarantees a point satisfying

Thus

Practice 4

The average slope is

Since

solve

Because \(c\in(0,2)\),

Practice 5

For \(x\ne y\), the Mean Value Theorem gives a point \(c\) between \(x\) and \(y\) such that

Taking absolute values,

Hence

Practice 6

The interval \([2,5]\) is compact. A continuous function on a compact set is uniformly continuous. Therefore, for sufficiently short subintervals, the oscillation of the function is uniformly small. Choosing a fine enough partition makes

arbitrarily small. By the Darboux criterion, \(f\) is Riemann integrable.

Practice 7

Every nonempty subinterval contains rational and irrational numbers. Therefore on every subinterval,

Since the interval has total length \(1\),

and

for every partition \(P\). Their difference is always \(3\), so the function is not Riemann integrable.

Practice 8

The integrand

is continuous. By the Fundamental Theorem of Calculus,

Practice 9

We have

Therefore,

Uniform convergence means

so the integral difference tends to zero.

Practice 10

Uniform convergence of the functions alone gives no corresponding control of their derivatives. Additional hypotheses are needed, typically uniform convergence of the derivatives together with convergence at one point.

For example,

converges uniformly to \(0\), but

does not converge in the required way.

What to Remember

Differentiability is defined by a limit:

Differentiability implies continuity, but continuity alone does not imply differentiability.

The Mean Value Theorem converts derivative information into global control:

for some \(c\in(a,b)\), provided its hypotheses hold.

Riemann integration is defined through stable limiting behavior of partition sums. The Darboux criterion is

for suitably chosen partitions.

The Fundamental Theorem connects the two processes:

for continuous \(f\), and

when \(F'=f\).

The governing habit of rigorous analysis is to identify the theorem, verify every hypothesis, and only then use the conclusion.

Where We Go Next

Riemann integration handles continuous functions and many familiar discontinuous functions, but modern analysis requires a broader framework for measuring sets, controlling exceptional points, and passing limits through integrals.

The next lesson, Measure and Measurable Functions, introduces sigma-algebras, measures, measurable sets, measurable functions, null sets, and the structural language needed for Lebesgue integration.