# Convergence to the derivative

1. Mar 31, 2008

### jostpuur

Let

$$f:]a,b[\to\mathbb{R}$$

be a continuously differentiable function, where $]a,b[\subset\mathbb{R}$ is some interval, and define

$$f_n:]a,b[\to\mathbb{R},\quad f_n(x)=\left\{\begin{array}{ll} &\frac{f(x+1/n)-f(x)}{1/n},\quad x\in ]a,b-1/n[\\ &\textrm{something continuous},\quad x\in [b-1/n, b[\\ \end{array}\right.$$

Clearly we have a point wise limit $f_n(x)\to f'(x)$ as $n\to\infty$, but how common it is that this convergence is uniform? Is there some well known theorem that says that the convergence is uniform under some assumptions?

Last edited: Apr 1, 2008