A continuous function on a closed interval is uniformly continuous
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 \(\varepsilon > 0\), there exists \(\delta > 0\), such that, for all \(x, y\in I\), \(|x - y| < \delta\) implies \(|f(x) - f(y)|< \varepsilon\). In a logical form, this condition is expressed as \[ \forall \varepsilon > 0, \exists \delta > 0, \forall x,y\in I ~ (|x-y| < \delta \implies |f(x) - f(y)| < \varepsilon).\label{eq:unifcont} \] Remark . Compare the above condition for uniform continuity with the condition for the continuous functi