### Differentiability

Due by Mon, 12 Mar 2018

1. Consider the function $f : \mathbb{R}^2 \to \mathbb{R}$ given by
• $f(0, 0) = 0$,
• $f(x, y) = 1$ if $y = x^2$ and $x \neq 0$,
• $f(x, y) = 0$ if $y \neq x^2$.

Show that

• (a) $f$ is not continuous at $(0, 0)$.
• (b) for every vector $v \in \mathbb{R}^2$, the directional derivative of $f$ at $(0, 0)$ in the direction $v$ exists.
2. Consider the function $f : \mathbb{R}^2 \to \mathbb{R}$ given by
• $f(0, 0) = 0$,
• $f(x, y) = x$ if $y = x^2$ and $x \neq 0$,
• $f(x, y) = 0$ if $y \neq x^2$.

Show that

• (a) $f$ is continuous at $(0, 0)$.
• (b) for every vector $v \in \mathbb{R}^2$, the directional derivative of $f$ at $(0, 0)$ in the direction $v$ exists.
• (c) The total derivative $Df((0, 0))$ of $f$ at $(0, 0)$ does not exist.
3. Consider the function $f : \mathbb{R}^2 \to \mathbb{R}$ given by
• $f(0, 0) = 0$,
• $f(x, y) = \frac{xy}{x^2 + y^2}$ if $(x, y) \neq (0, 0)$,

Show that

• (a) $f$ is not continuous at $(0, 0)$.
• (b) For every $(x, y)\in \mathbb{R^2}$, the partial derivatives $\frac{\partial f}{\partial x}$ and $\frac{\partial f}{\partial y}$ exist.
• (c) The partial derivatives $\frac{\partial f}{\partial x}$ and $\frac{\partial f}{\partial y}$ are not continuous functions.
4. Consider the function $f : \mathbb{R}^2 \to \mathbb{R}$ given by
• $f(0, 0) = 0$,
• $f(x, y) = \frac{xy}{x^{4/3} + y^{4/3}}$ if $(x, y) \neq (0, 0)$,

Show that

• (a) $f$ is continuous at all $(x, y)\in \mathbb{R}^2$.
• (b) For every $(x, y)\in \mathbb{R^2}$, the partial derivatives $\frac{\partial f}{\partial x}$ and $\frac{\partial f}{\partial y}$ exist.
• (c) The partial derivatives $\frac{\partial f}{\partial x}$ and $\frac{\partial f}{\partial y}$ are not continuous functions.
• (d) The total derivative $Df((0, 0))$ of $f$ at $(0, 0)$ does not exist.