Proof that if $f:\mathbb{R^n}\to \mathbb{R^m}$ is differentiable then it's continuous

Question

I've read the proof given below and am confused about the proof. I know I want to show that $f(p+h)-f(p) \to 0$ as $h \to 0$. Can someone explain their process? An alternative proof would be helpful. And why couldn't we use the proof used in the single variable case where we write $\lim_{h\to 0}f(p+h)-f(p)= \dfrac{\big(f(p+h)-f(p)\big)|h|}{|h|} \Longrightarrow f'(p)|h| \Longrightarrow f'(p)0=0$?

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Answer 1

Here's a much clearer proof!

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