# Thread: Proving continuity of function

1. ## Proving continuity of function

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

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

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

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