Condition for Lipschitz Condition to be Satisfied

Theorem
Let $f$ be a real function.

Then $f$ satisfies the Lipschitz condition on a closed real interval $\left[{a \,.\,.\, b}\right]$ if:
 * $\forall y \in \left[{a \,.\,.\, b}\right]: \exists A \in \R: \left|{\phi^{\prime} \left({y}\right)}\right| \le A$

Proof
Integrating both sides of $\left|{\phi^{\prime} \left({y}\right)}\right| \le A$ gives us:

On the interval $\left[{a \,.\,.\, b}\right]$ it follows that $\left|{\phi \left({y}\right)}\right|$ is bounded by the greater of $A a$ and $A b$.

Hence the result.