---
tag: 0CS42KEM5C
kind: statement
role: theorem
name: Uniform convergence of derivatives allows term-by-term differentiation
msc: [40A30, 26A24]
uses: [08FPABDQ4C, 0BARQY52N5, 01XAY17QA3]
sources:
  - url: https://en.wikipedia.org/wiki/Uniform_convergence
    locator: To differentiability
---

## Statement

Let $a < b$, and let $f_n \colon [a, b] \to \mathbb{R}$ be differentiable for each $n$.
Assume that $(f_n(x_0))$ converges for some $x_0 \in [a, b]$, and that $(f_n')$ converges [[uniformly|08FPABDQ4C]] on $[a, b]$ to a function $g$.
Then $(f_n)$ converges uniformly on $[a, b]$ to a differentiable function $f$, and $f' = g$.

## Remarks

The [[mean value theorem|0BARQY52N5]] applied to $f_n - f_m$ shows that $(f_n)$ is [[uniformly Cauchy|01XAY17QA3]].
Uniform convergence of $f_n$ alone is not enough. The functions $\frac{\sin(n^2 x)}{n}$ tend to $0$ uniformly, but their derivatives $n \cos(n^2 x)$ do not converge.
Mathlib states a version as `hasDerivAt_of_tendstoUniformly`.

## Explanations

### L1 human

Derivatives are sensitive to small wiggles, so uniform convergence of functions does not control them.
We need uniform convergence of the derivatives themselves.