2002-12.md

course	course_year	question_number	tags	title	year
Analysis II	IB	12	IB 2002 Analysis II	1.I.1E	2002

Suppose that for each $n=1,2, \ldots$, the function $f_{n}: \mathbb{R} \rightarrow \mathbb{R}$ is uniformly continuous on $\mathbb{R}$.

(a) If $f_{n} \rightarrow f$ pointwise on $\mathbb{R}$ is $f$ necessarily continuous on $\mathbb{R}$ ?

(b) If $f_{n} \rightarrow f$ uniformly on $\mathbb{R}$ is $f$ necessarily continuous on $\mathbb{R}$ ?

In each case, give a proof or a counter-example (with justification).