2007-21.md

course	course_year	question_number	tags	title	year
Analysis II	IB	21	IB 2007 Analysis II	2.II.13H	2007

Show that the limit of a uniformly convergent sequence of real valued continuous functions on $[0,1]$ is continuous on $[0,1]$.

Let $f_{n}$ be a sequence of continuous functions on $[0,1]$ which converge point-wise to a continuous function. Suppose also that the integrals $\int_{0}^{1} f_{n}(x) d x$ converge to $\int_{0}^{1} f(x) d x$. Must the functions $f_{n}$ converge uniformly to $f ?$ Prove or give a counterexample.

Let $f_{n}$ be a sequence of continuous functions on $[0,1]$ which converge point-wise to a function $f$. Suppose that $f$ is integrable and that the integrals $\int_{0}^{1} f_{n}(x) d x$ converge to $\int_{0}^{1} f(x) d x$. Is the limit $f$ necessarily continuous? Prove or give a counterexample.