I need help with prove "If the partial derivatives are bounded then the function is Lipschitz".

Question

Can anyone help me to prove the next statements?

1. Let $f:\mathbb R^n$$\to$$\mathbb R$ such that the partial derivatives $\frac{\partial f}{\partial x_{i}}$ exists at each point and $E$ $\subseteq$ $\mathbb R^n$. So, if the partial derivatives are bounded in $E$, then $f$ Satisfies the Lipschitz condition:

   $|f(x)-f(y)|$$\le$$M|x-y|$

2. Let $E$ $\subseteq$ $\mathbb R^n$ open subset and $f: E$$\to$$\mathbb R$ Differentiability and Continuous then $f$ Satisfies the **Lipschitz condition*

Thank you.

Answer 1

It's not true, without further assumption on the set $E$ - for example, if $E=\{x,y\}$ contains exactly two points there is no way to obtain a Lipschitz bound from knowledge of the gradient of the function at just those two points.

On the other hand, if $E$ is path-connected then you can prove the result by picking a path from $x$ to $y$ and integrating the gradient of $f$ along this path, using the triangle inequality for integrals to get the Lipschitz bound. To be more precise, you will need to know that the length of the path is bounded by a multiple of the distance from $x$ to $y$, which can be arranged as long as $E$ is not too pathological. For instance, if $E$ is convex you can just pick the straight line path.