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.