2007-22.md

course	course_year	question_number	tags	title	year
Analysis II	IB	22	IB 2007 Analysis II	$3 . \mathrm{I} . 3 \mathrm{H}$	2007

Define uniform continuity for a real-valued function on an interval in the real line. Is a uniformly continuous function on the real line necessarily bounded?

Which of the following functions are uniformly continuous on the real line?

(i) $f(x)=x \sin x$,

(ii) $f(x)=e^{-x^{4}}$.

Justify your answers.