Let $\{f_{n}:[a,b]\to\mathbb{R}\}$ be a sequence of Riemann integrable functions which converges pointwise to a Riemann integrable function $f:[a,b]\to\mathbb{R}$. Now, suppose that $\{f_{n}\}$ converges uniformly to $f$ on $(a,b]$.

Q1) Under this hypothesis, is it possible to show that $\{f_{n}\}$ converges uniformly to $f$ on $[a,b]$? (without notions of measure theory)

Q2) More generally, under what conditions, the uniform convergence extends $(a,b]$ to $[a,b]$? I wonder if there is a simple case.

Please, let me know some advice or any reference. Thank you!

Hint:

For all $\epsilon > 0$ there exists $N_1,N_2 \in \mathbb{N}$ such that

$\sup_{x \in (a,b]} |f_n(x) -f(x)| < \epsilon$ for $n > N_1,$

and

$|f_n(a)-f(a)| <\epsilon$ for $n > N_2$

October 10, 2019 02:47 AM

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