Brunei Math Club

The notion of  uniform continuity  is a ``stronger'' version of (simple) continuity. If a function is uniformly continuous, it is continuous, but the converse does not generally hold (that is, a continuous function may not be uniformly continuous). However, if we restrict a continuous function on a closed interval, it is always uniformly continuous. Definition (Uniform continuity) The function $f(x)$ on an interval $I$ is said to be  uniformly continuous  on $I$ if it satisfies the following condition. For any $\epsilon >0$, there exists $\delta >0$, such that, for all $x,y\in I$, $|x-y|<\delta$ implies $|f(x)-f(y)|<\epsilon$.   In a logical form, this condition is expressed as  $$\mathrm{\forall }\epsilon >0,\mathrm{\exists }\delta >0,\mathrm{\forall }x,y\in I(|x-y|<\delta ⟹|f(x)-f(y)|<\epsilon ).$$ Remark . Compare the above condition for uniform continuity with the condition...