The Baire Category Theorem with an Application to Real Analysis

Show that $X$ is nowhere dense in $M$ if and only if its closure has empty interior, $(\bar{X})^{\circ }=\emptyset$.

Show that any finite subset of $\mathbb {R}$ is nowhere dense in $\mathbb {R}$. Give examples to show that a countable set can be nowhere dense in $\mathbb {R}$, but that this is not always the case.

Let $f(x)=\sin \left(\frac{1}{x}\right)$ for $x \neq 0$, and $f(0)=0$. Compute $\omega_{f}(0)$. How does this tie in with what you know about the continuity of this function?

First category sets are in some sense "small," while second category sets are "large." Another notion of set size that we have discussed is "measure zero": "Small" sets are of measure zero, while "large" sets are not. Is there a connection between these two ways of describing the size of a set? The answer is no. Please show this by doing two things:

Prove that the function defined on $(0,1)$ by

is continuous at each irrational point of $(0,1)$ and discontinuous at each rational point of $(0,1)$.

The goal of this exercise is to show that "most" continuous functions are nowhere differentiable. Let $E_{n}$ denote the set of all $f \in C([0,1])$ for which there exists an $x_{f} \in [0,1]$ satisfying

for every $x \in [0,1]$.