Let $f$ be continuous on [a,b].
Then $f$ is uniformly continuous on [a,b] and there exists $\delta >0$ such that
$|f(s)-f(t)|<\epsilon$ if $|s-t|<\delta$.
Let P={$x_0,x_1,...,x_n$} is a partition of [a,b] with $x_i-x_{i-1}<\delta$ for all i.
If $x_{i-1} \le t \le x_i$ then
$|f(t)|\le |f(x_i)|+\epsilon$
Where does this last inequality come from?