Differentiability and the Mean Value Theorem

Differentiability is the mathematical heart of calculus, providing a precise tool for analyzing instantaneous change. While continuity ensures a function has no breaks, differentiability imposes the stricter—and far more powerful—condition that a function has a well-defined, linear best approximation at a point. This crucial distinction builds to the Mean Value Theorem, a profound result that connects the local concept of the derivative to the global behavior of a function, forming the bedrock for major applications like testing monotonicity, justifying L'Hôpital's Rule, and establishing the accuracy of polynomial approximations.

Differentiability: More Than Just Smoothness

A function $f$ is said to be differentiable at a point $x=a$ if the derivative $f^{\prime }(a)$ exists. Formally, this means the following limit exists: $$f^{\prime }(a)=\underset{h\to 0}{\lim }\frac{f(a+h)-f(a)}{h}.$$ Geometrically, this limit defines the slope of the unique tangent line to the graph of $f$ at the point $(a,f(a))$. This is a significantly stronger condition than continuity.

Every differentiable function is continuous, but the converse is false. Continuity only requires that the function's value approaches $f(a)$ as $x$ approaches $a$. Differentiability requires that the function's rate of change approaches a fixed value, which implies the graph must not only be unbroken but also "smooth" without any sharp corners, cusps, or vertical tangents at $a$. For example, $f(x)=|x|$ is continuous at $x=0$ but not differentiable there, as the graph has a sharp corner. The limit from the left yields a slope of $-1$, while from the right it yields $+1$, so the two-sided limit defining $f^{\prime }(0)$ does not exist.

The Mean Value Theorem: The Bridge Between Local and Global

The Mean Value Theorem (MVT) is a cornerstone of differential calculus. It states: If a function $f$ is continuous on the closed interval $[a,b]$ and differentiable on the open interval $(a,b)$, then there exists at least one point $c$ in $(a,b)$ such that $$f^{\prime }(c)=\frac{f(b)-f(a)}{b-a}.$$

In plain language, under the right conditions, there is at least one interior point where the instantaneous rate of change (the derivative) equals the average rate of change over the entire interval. Visually, on the graph of $f$, there is a point where the tangent line is parallel to the secant line connecting the endpoints $(a,f(a))$ and $(b,f(b))$.

The proof relies on a clever auxiliary function and Rolle's Theorem, a special case of the MVT where $f(a)=f(b)$. One defines $g(x)=f(x)-\left[\frac{f(b)-f(a)}{b-a}(x-a)+f(a)\right]$. This new function $g(x)$ subtracts the secant line from $f(x)$, guaranteeing $g(a)=g(b)=0$. Applying Rolle's Theorem to $g(x)$ on $[a,b]$ yields a point $c$ where $g^{\prime }(c)=0$. Since $g^{\prime }(c)=f^{\prime }(c)-\frac{f(b)-f(a)}{b-a}$, setting this to zero gives the desired result.

Consequences I: Monotonicity and L'Hôpital's Rule

The MVT provides powerful, simple tests for how a function behaves over an interval. A key application is establishing monotonicity criteria. If $f^{\prime }(x)>0$ for all $x$ in an interval $I$, then $f$ is strictly increasing on $I$. The proof is direct: take any two points $x_1<x_2$ in $I$. By the MVT on $[x_1,x_2]$, there exists a $c$ such that $f(x_2)-f(x_1)=f^{\prime }(c)(x_2-x_1)$. Since $f^{\prime }(c)>0$ and $(x_2-x_1)>0$, we have $f(x_2)>f(x_1)$. Similarly, $f^{\prime }(x)<0$ implies the function is strictly decreasing.

The MVT also underpins the justification for L'Hôpital's Rule for the indeterminate form $0/0$. The core of the proof involves applying the Cauchy Mean Value Theorem, a generalization of the MVT to two functions. For functions $f$ and $g$ satisfying the appropriate conditions, the Cauchy MVT guarantees a point $c$ where $\frac{f^{\prime }(c)}{g^{\prime }(c)}=\frac{f(b)-f(a)}{g(b)-g(a)}$. By strategically choosing an interval shrinking toward the point of the limit, one can show that the limit of the ratio of functions equals the limit of the ratio of their derivatives, provided the latter limit exists.

Consequences II: Taylor's Theorem with Remainder

While the derivative gives a linear (first-order) approximation of a function, Taylor's Theorem extends this idea to polynomial approximations of any degree. It tells us how well a function can be approximated by its Taylor polynomial and, crucially, provides an exact formula for the error, known as the remainder.

One common form is the Lagrange Remainder. If $f$ is differentiable $n+1$ times on an interval containing $a$, then for any $x$ in that interval, $$f(x)=\underset{\text{Taylor polynomial }{P}_n(x)}{\underbrace{\sum _{k=0}^n\frac{f^{(k)}(a)}{k!}(x-a{)}^k}}+\underset{\text{Lagrange remainder }{R}_n(x)}{\underbrace{\frac{f^{(n+1)}(c)}{(n+1)!}(x-a{)}^{n+1}}},$$ where $c$ is some number between $a$ and $x$.

The proof is an elegant repeated application of the Mean Value Theorem. One defines an auxiliary function and applies Rolle's Theorem (or the MVT) repeatedly, essentially "peeling off" successive polynomial terms until the familiar MVT form of the remainder emerges. This theorem is foundational because it transforms approximation into an exact science, allowing us to control the error $R_n(x)$ by bounding the $(n+1)$-th derivative.

Common Pitfalls

1. Assuming Differentiability Implies Continuous Derivatives: A function can be differentiable everywhere, yet its derivative can be discontinuous. A classic example is $f(x)=x^2\sin (1/x)$ for $x\ne 0$ and $f(0)=0$. This function is differentiable at $0$ (with $f^{\prime }(0)=0$), but $f^{\prime }(x)$ oscillates wildly as $x$ approaches $0$, making $f^{\prime }$ discontinuous at $0$. Differentiability only guarantees the function's continuity, not the derivative's.

1. Misapplying the Mean Value Theorem's Hypotheses: The MVT requires both continuity on $[a,b]$ and differentiability on $(a,b)$. If either condition fails, the conclusion may not hold. For instance, $f(x)=|x|$ on $[-1,1]$ is continuous but not differentiable at $0$, which is inside the interval. The average rate of change is $0$, but there is no point $c$ where $f^{\prime }(c)=0$, as the derivative is either $-1$, $+1$, or does not exist.

1. Confusing the "c" in the MVT and Taylor's Theorem: The point $c$ guaranteed by these theorems is known to exist but its exact value is generally unknown and depends on the function and the interval. It is a theoretical device for writing exact formulas, not a quantity you are typically meant to compute explicitly.

1. Overlooking the Remainder in Taylor Series: It's a major error to write $f(x)=P_n(x)$ without considering the remainder $R_n(x)$. Taylor's Theorem states equality only when the remainder is included. Ignoring it can lead to incorrect conclusions about the accuracy of a polynomial approximation, especially for values of $x$ far from the center $a$.

Summary

• Differentiability is a stronger condition than continuity, requiring a function to have a well-defined tangent line (instantaneous rate of change) at a point, which implies the function has no sharp corners or cusps there.
• The Mean Value Theorem rigorously connects a function's average rate of change over an interval to its instantaneous rate of change at some interior point, provided the function is continuous on the closed interval and differentiable on its interior.
• A direct consequence of the MVT is a clean test for monotonicity: a positive derivative on an interval implies the function is strictly increasing there, and a negative derivative implies it is strictly decreasing.
• The theoretical justification for L'Hôpital's Rule (for the $0/0$ form) relies on the Cauchy Mean Value Theorem, a generalization of the standard MVT.
• Taylor's Theorem with Remainder uses repeated applications of the MVT to provide an exact formula for approximating a function with a polynomial, complete with a precise expression for the error, enabling controlled local approximation.