2012-2.md

course	course_year	question_number	tags	title	year
Analysis II	IB	2	IB 2012 Analysis II	Paper 2, Section I, E	2012

Let $f: \mathbb{R}^{2} \rightarrow \mathbb{R}$ be a function. What does it mean to say that $f$ is differentiable at a point $(x, y) \in \mathbb{R}^{2} ?$ Prove directly from this definition, that if $f$ is differentiable at $(x, y)$, then $f$ is continuous at $(x, y)$.

Let $f: \mathbb{R}^{2} \rightarrow \mathbb{R}$ be the function:

$$f(x, y)= \begin{cases}x^{2}+y^{2} & \text { if } x \text { and } y \text { are rational } \ 0 & \text { otherwise. }\end{cases}$$

For which points $(x, y) \in \mathbb{R}^{2}$ is $f$ differentiable? Justify your answer.