Differentiability
Due by Mon, 12 Mar 2018
- 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.
- 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.
- 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.
- 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.