Ir al contenido principal

Due February 7th

Problem 1

If $f$ is differentiable at $z$, then $f$ is continuous at $z$.

Problem 2

Let $T:\mathbb C \to \mathbb C$ be $\mathbb R$-linear.
1. $T$ is $\mathbb C$-linear if and only if $T(iz) = iTz$ for all $z\in\mathbb C$.
2. If $T$ has matrix $\begin{pmatrix}a & b\\c & d\end{pmatrix}$, then $T$ is $\mathbb C$-linear if and only if $a = d$ and $b = -c$.

Problem 3

Define $f:\mathbb C\to\mathbb C$ by
$f(x+iy) = \begin{cases} 0 & x=0\\ 0 & y=0\\ 1 & \text{otherwise.}\end{cases}$
Then $f$ satisfies the Cauchy-Riemann equations at the origin, but it is not differentiable at the origin.

Problem 4

Define $f:\mathbb C\to\mathbb C$ by
$f(z) = \begin{cases} |z|^2\sin(1/|z|) & z\not=0\\ 0 & z=0.\end{cases}$
Then $f$ is differentiable at the origin, but the partial derivative $\partial_x u$ is not continuous at the origin.

Problem 5

Let $a\in[0,1]$. Then there exist sequences $x_n, y_n$ of positive numbers converging to 0 such that $x_n^{y_n}\to a$.