Let $f$ be a continous differentiable function on $\mathbb R$. Let $f_n$ be a sequence of functions $f_n = n(f(x+ \frac{1}{n})-f(x))$. Then
(a) $f_n$ converges uniformly on $\mathbb R$
(b) $f_n$ converges on $\mathbb R$, but not necessarily uniformly.
(c) $f_n$ converges to the derivative of $f$ uniformly on$[0,1]$
(d) there is no guarantee that $f_n$ converges on any open interval.
We know that $f'(x) =\lim_{h \to 0} \frac{f(x+h) - f(x)}{h} = \lim_{n \to\infty} \frac{f(x+\frac{1}{n}) - f(x)}{\frac{1}{n}}=\lim_{n \to \infty} f_n(x)$. Thus $f_n$ converges pointwise to $f'$
Please tell me how to check the uniform convergence. Any help would be appreciated. Thank you.
A hint: You can write $f_n$ in the form $$f_n(x)=\int_0^1 f'\left(x+{\tau\over n}\right)\>d\tau\ .$$