# Example of sequence of functions

$$f$$ is continuously differentiable function on $$\Bbb R$$

Define $$f_n(x)=\dfrac{f(x+1/n)-f(x)}{1/n}$$

I want example of function such that $$f_n\to f'$$ but not uniformly on $$\Bbb R$$

I thought that above $$f_n$$ converges uniformly but this is not correct. Please can anyone help me to find example

where is my intitution going wrong?

any help will be appreciated

• Doesn't $f(x)=x^3$ work? – yoyo Dec 7 '18 at 13:44
• Because that $f_{\frac{1}{n}}(x)=3x^2+\frac{1}{n^2}+\frac{3x}{n^2}$. And the term $\frac{3x}{n^2}$ is not uniformly convergent. – yoyo Dec 7 '18 at 13:47
• If the derivative is uniformly continuous then this convergence must be uniform. Since you specified that $f$ has a continuous derivative, the counterexample must be where that continuity is not uniform. @yoyo gives a good example. – RRL Dec 7 '18 at 14:11
• This is not a duplicate of the suggested target. – RRL Dec 7 '18 at 14:12
• @RRL Thank you~!! – yoyo Dec 7 '18 at 14:42

Let $$f(x)=x^3$$. Then it will be an example.
Since $$f_{\frac{1}{n}}(x)=3x^2+\frac{1}{n^2}+\frac{3x}{n}$$. But the term $$\frac{3x}{n}$$ is not convergent uniformly to $$0$$ as $$n$$ goes to $$\infty$$.