Basic Analysis I — Real Analysis · Problem 4.2.4

The Derivative — Jiri Lebl

Mathematics — Real Analysis · derivative-sequence-approximation

Problem

Suppose f:[a,b]→Rf : [a,b] \to \mathbb{R} is differentiable and c∈[a,b]c \in [a,b]. Show there exists a sequence {xn}n=1∞\{x_n\}_{n=1}^\infty converging to cc, xn≠cx_n \neq c for all nn, such that f′(c)=lim⁡n→∞f′(xn).f'(c) = \lim_{n \to \infty} f'(x_n). Do note this does not imply that f′f' is continuous (why?).
Intuitive explanation

The derivative f′(c)f'(c) is defined as a limit of difference quotients, not as a pointwise value we can directly access. The key insight is to construct a sequence of points where the difference quotient f(xn)−f(c)xn−c\frac{f(x_n) - f(c)}{x_n - c} approximates f′(c)f'(c) increasingly well, then use the Mean Value Theorem to find corresponding points xnx_n in the intervals between cc and these approximation points where f′(xn)f'(x_n) equals those difference quotients. This works because MVT guarantees a derivative value matching any difference quotient we need.

Intuitive explanation

The derivative f′(c)f'(c) is defined as a limit of difference quotients, not as a pointwise value we can directly access. The key insight is to construct a sequence of points where the difference quotient f(xn)−f(c)xn−c\frac{f(x_n) - f(c)}{x_n - c} approximates f′(c)f'(c) increasingly well, then use the Mean Value Theorem to find corresponding points xnx_n in the intervals between cc and these approximation points where f′(xn)f'(x_n) equals those difference quotients. This works because MVT guarantees a derivative value matching any difference quotient we need.

Related theory

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