Notes - Analysis II HT23, Pathological examples
Flashcards
Can you give an example of a function that is both continuous at an infinite number of points and also discontinuous at an infinite number of points?
Thomae’s function
\[T(x) = \begin{cases} \frac{1}{q} &\text{if }x=\frac p q \text{, with } p\in\mathbb Z \text{ } \text{ and } q \in \mathbb N \text{ coprime} \\\\ 0 &\text{if } x \text{ is irrational} \end{cases}\]Can you give an example of a function that is continuous nowhere?
\[f(x) = \begin{cases}
1 &x \in \mathbb Q \\\\
0 &x \notin \mathbb Q
\end{cases}\]
Can you give an example of a continuous function that is nowhere differentiable?
Weierstrass’ function
\[f(x) = \sum^\infty_{k=0} 2^{-k} \cos(10^k \cdot 2\pi x)\]Can you give an example of a function whose Taylor series doesn’t converge to the function anywhere?
\[f(x) = e^{1/x}\]