From my textbook, Measure and Integral, a function $f$ defined on $[a, b]$ is said to satisfy a Lipschitz condition on $[a, b]$ or to be a Lipschitz function on $[a, b]$, if there is a constant $C$ such that $$\left|f(x)-f(y)\right|\le C|x-y| \quad \text{for all } x, y \in [a, b]$$
a Lipschitz function is of bounded variation.
if $f$ has continuous derivative on $[a, b]$, then $f$ satisfies a Lipschitz condition on $[a, b]$.
I accepted the definition of a Lipschitz condition and understood the two theorems.
However, my professor taught us if $|f'(x)|\le C$ for all $x\in [a, b]$, then $f$ is a Lipschitz function.
I do not understand how come the second theorem changes into this proposition.
Please some people tell me this?