2014-4.md

course	course_year	question_number	tags	title	year
Analysis II	IB	4	IB 2014 Analysis II	Paper 4, Section II, F	2014

Let $U \subset \mathbb{R}^{2}$ be an open set. Define what it means for a function $f: U \rightarrow \mathbb{R}$ to be differentiable at a point $\left(x_{0}, y_{0}\right) \in U$.

Prove that if the partial derivatives $D_{1} f$ and $D_{2} f$ exist on $U$ and are continuous at $\left(x_{0}, y_{0}\right)$, then $f$ is differentiable at $\left(x_{0}, y_{0}\right)$.

If $f$ is differentiable on $U$ must $D_{1} f, D_{2} f$ be continuous at $\left(x_{0}, y_{0}\right) ?$ Give a proof or counterexample as appropriate.

The function $h: \mathbb{R}^{2} \rightarrow \mathbb{R}$ is defined by

$$h(x, y)=x y \sin (1 / x) \quad \text { for } x \neq 0, \quad h(0, y)=0$$

Determine all the points $(x, y)$ at which $h$ is differentiable.