How does one show that given that $g\in C^1(a,b)$, the sequence of functions $$ g_n=n\left(g\left(x+{1\over n}\right)-g(x)\right) $$ converges uniformly on all closed intervals in $(a,b)$? I assume the limit function is $g'(x)$.
Uniform convergence and differentiation
2
$\begingroup$
real-analysis
analysis