# Continuity

## 1 #complex/qual/work

Is the following function continuous, differentiable, continuously differentiable? \begin{align*} f: {\mathbb{R}}^2 &\to {\mathbb{R}}\\ f(x, y) &= \begin{cases} {xy \over \sqrt{x^2 + y^2}} & (x, y) \neq (0, 0) \\ 0 & \text{else}. \end{cases} \end{align*}

## ? #complex/qual/work

Show that $$f(z) = z^2$$ is uniformly continuous in any open disk $$|z| < R$$, where $$R>0$$ is fixed, but it is not uniformly continuous on $$\mathbb C$$.

## 6 #complex/qual/work

Let $$F:{\mathbb{R}}^2\to {\mathbb{R}}$$ be continuously differentiable with $$F(0, 0) = 0$$ and $${\left\lVert {\nabla F(0, 0)} \right\rVert} < 1$$.

Prove that there is some real number $$r> 0$$ such that $${\left\lvert {F(x, y)} \right\rvert} < r$$ whenever $${\left\lVert {(x, y)} \right\rVert} < r$$.

## 2 Multivariable derivatives #complex/qual/work

• Complete this definition: “$$f: {\mathbb{R}}^n\to {\mathbb{R}}^m$$ is real-differentiable a point $$p\in {\mathbb{R}}^n$$ iff there exists a linear transformation…”

• #complex/qual/work Give an example of a function $$f:{\mathbb{R}}^2\to {\mathbb{R}}$$ whose first-order partial derivatives exist everywhere but $$f$$ is not differentiable at $$(0, 0)$$.

• #complex/qual/work Give an example of a function $$f: {\mathbb{R}}^2 \to {\mathbb{R}}$$ which is real-differentiable everywhere but nowhere complex-differentiable.

# Implicit/Inverse Function Theorems

## 3 #complex/qual/work

Let $$f:{\mathbb{R}}^2\to {\mathbb{R}}$$.

• #complex/qual/work Define in terms of linear transformations what it means for $$f$$ to be differentiable at a point $$(a, b) \in {\mathbb{R}}^2$$.

• #complex/qual/work State a version of the inverse function theorem in this setting.

• #complex/qual/work Identify $${\mathbb{R}}^2$$ with $${\mathbb{C}}$$ and give a necessary and sufficient condition for a real-differentiable function at $$(a, b)$$ to be complex differentiable at the point $$a+ib$$.

## 5 #complex/qual/work

Let $$P = (1, 3) \in {\mathbb{R}}^2$$ and define \begin{align*} f(s, t) \coloneqq ps^3 -6st + t^2 .\end{align*}

• State the conclusion of the implicit function theorem concerning $$f(s, t) = 0$$ when $$f$$ is considered a function $${\mathbb{R}}^2\to{\mathbb{R}}$$.

• State the above conclusion when $$f$$ is considered a function $${\mathbb{C}}^2\to {\mathbb{C}}$$.

• Use the implicit function theorem for a function $${\mathbb{R}}\times{\mathbb{R}}^2 \to {\mathbb{R}}^2$$ to prove (b).

There are various approaches: using the definition of the complex derivative, the Cauchy-Riemann equations, considering total derivatives, etc.

## 7 #complex/qual/work

State the most general version of the implicit function theorem for real functions and outline how it can be proved using the inverse function theorem.

# Complex Differentiability

## 4 #complex/qual/work

Let $$f = u+iv$$ be complex-differentiable with continuous partial derivatives at a point $$z = re^{i\theta}$$ with $$r\neq 0$$. Show that \begin{align*} {\frac{\partial u}{\partial r}\,} = {1\over r}{\frac{\partial v}{\partial \theta}\,} \qquad {\frac{\partial v}{\partial r}\,} = -{1\over r}{\frac{\partial u}{\partial \theta}\,} .\end{align*}

## Tie’s Extra Questions: Fall 2016

Let $$u(x,y)$$ be harmonic and have continuous partial derivatives of order three in an open disc of radius $$R>0$$.

• Let two points $$(a,b), (x,y)$$ in this disk be given. Show that the following integral is independent of the path in this disk joining these points: \begin{align*} v(x,y) = \int_{a,b}^{x,y} ( -\frac{\partial u}{\partial y}dx + \frac{\partial u}{\partial x}dy) .\end{align*}

• In parts:

• Prove that $$u(x,y)+i v(x,y)$$ is an analytic function in this disc.
• Prove that $$v(x,y)$$ is harmonic in this disc.

## Tie’s Questions, Spring 2014: Polar Cauchy-Riemann #complex/qual/work

Let $$f=u+iv$$ be differentiable (i.e. $$f'(z)$$ exists) with continuous partial derivatives at a point $$z=re^{i\theta}$$, $$r\not= 0$$. Show that \begin{align*} \frac{\partial u}{\partial r}=\frac{1}{r}\frac{\partial v}{\partial \theta},\quad \frac{\partial v}{\partial r}=-\frac{1}{r}\frac{\partial u}{\partial \theta} .\end{align*}

## ? #complex/qual/work

• Show that the function $$u=u(x,y)$$ given by \begin{align*} u(x,y)=\frac{e^{ny}-e^{-ny}}{2n^2}\sin nx\quad \text{for}\ n\in {\mathbf N} \end{align*} is the solution on $$D=\{(x,y)\ | x^2+y^2<1\}$$ of the Cauchy problem for the Laplace equation \begin{align*}\frac{\partial ^2u}{\partial x^2}+\frac{\partial ^2u}{\partial y^2}=0,\quad u(x,0)=0,\quad \frac{\partial u}{\partial y}(x,0)=\frac{\sin nx}{n}.\end{align*}

• Show that there exist points $$(x,y)\in D$$ such that $$\displaystyle{\limsup_{n\to\infty} |u(x,y)|=\infty}$$.

#complex/qual/work