Proving continuity of function

**JG89** · June 16th 2010, 11:29 AM

I would just like someone to check this proof for me. I am just learning this material so I would like to make sure I am doing the proofs properly.

Question: Show that if $A \subset R^n$ and $f: A \rightarrow R$ and if the partial derivatives $\partial f_{x_i}$ exist and are bounded in a neighborhood of $\vec{a} \in R^n$ , then f is continuous at $\vec{a}$

Proof: We have to show that $|f(\vec{a} + t\vec{e_i}) - f(\vec{a})| \rightarrow 0$ as $t \rightarrow 0$ where $t$ is a real number and $\vec{e_i}$ is any basis vector for $R^n$ .

Let $M$ be an upper bound for all the partial derivatives of f, in some neighborhood of $\vec{a}$ . Observe that $0 \le \lim_{t \rightarrow 0} |f(\vec{a} + t\vec{e_i}) - f(\vec{a})| = \lim_{t \rightarrow 0} |t| |\frac{f(\vec{a} + t\vec{e_i}) - f(\vec{a})}{t}| = \lim_{t \rightarrow 0} |t| \lim_{t \rightarrow 0} |\frac{|f(\vec{a} + t\vec{e_i}) - f(\vec{a})}{t}|$ $= \lim_{t \rightarrow 0} |t| \partial f_{x_i} \le \lim_{t \rightarrow 0} |t| M = 0$ and so $|f(\vec{a} + t\vec{e_i}) - f(\vec{a})| \rightarrow 0$ as desired. QED

**maddas** · June 16th 2010, 01:02 PM

Did you use the fact that the derivatives exist and are bounded in a neighborhood of $a$ ?

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