# Extending the interval of uniform convergence to end point

by Primavera   Last Updated October 10, 2019 03:20 AM - source

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.

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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 $$n > N_2$$

RRL
October 10, 2019 02:47 AM