As stated in my notes:
Remark: Suppose $f: E \to \mathbb{R}$, $E \subseteq \mathbb{R}^n$, and $p \in E$. Also, suppose that $D_if$ exists in some neighborhood of $p$, say, $N(p, h)$ where $h>0$. If all partial derivatives of $f$ are bounded, then $f$ is continuous on $E$.
I found a sketch of the proof here. I'm wondering if I can adapt this proof as follows:
$f(x_1+h_1,…,x_n+h_n)-f(x_1,…,x_n)=f(x_1+h_1,…,x_n+h_n)-f(x_1,x_2+h_2,…,x_n+h_n)-…-f(x_1,x_2,…,x_{n-1}+h_{n-1},x_n+h_n)-f(x_1,…,x_{n-1},x_n+h_n)-f(x_1,…,x_n)$
However, I'm not sure how to apply the contraction principle to finish off the proof. Is there a more efficient way to prove the above remark?
Best Answer
The proof is a combination of two facts:
A function of one real variable with a bounded derivative is Lipschitz.
Let $Q\subset \mathbb R^n$ be a cube aligned to coordinate axes. If a function $f:Q\to\mathbb R$ is Lipschitz in each variable separately, then it is Lipschitz.
The proof of 2 involves a telescoping sum such as $$\begin{split} f(x,y,z)-f(x',y',z')&= f(x,y,z)-f(x',y,z) \\ & + f(x',y,z)-f(x',y',z)\\&+f(x',y',z)-f(x',y',z') \end{split}$$