Math test

Continuity( \newcommand\RR{\mathbf R} )

Let ( f \colon \RR \to \RR ) be any function. We say that ( f ) is continuous at ( x \in \RR ) if, for every ( \epsilon > 0 ) there exists ( \delta > 0 ) such that [ | x' - x | < \delta \Longrightarrow | f(x') - f(x) | < \epsilon. ] We say that ( f ) is continuous if for every ( x \in \RR ) and ( \epsilon > 0 ), we can find a ( \delta > 0 ) to satisfy the above condition. We say, further, that ( f ) is uniformly continuous if for every ( \epsilon > 0 ), there exists a ( \delta > 0 ), such that for all ( x \in \RR ) the above holds.

Clearly, uniform continuity implies continuity.