Condition for Lipschitz Condition to be Satisfied

Theorem
Let $\phi$ be a real function.

Then $\phi$ satisfies the Lipschitz condition on a closed real interval $\closedint a b$ if:
 * $\forall y \in \closedint a b: \exists A \in \R: \size {\map {\phi'} y} \le A$

Proof
Integrating both sides of $\size {\map {\phi'} y} \le A$ gives us:

On the interval $\closedint a b$ it follows that $\size {\map \phi y}$ is bounded by the greater of $A a$ and $A b$.

Hence the result.