Example of sequence of functions

Question

$f$ 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

Answer 1

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 $\infty$.