If $f:X \rightarrow Y$ is continuous and a sequence $\{x_n\}$ in $X$ converges to $x \in X$, show that the sequence $\{f(x_n)\}$ in $Y$ converges to $f(x)$.

Let $E$ be a non-empty proper subset of the topological space $X$. Consider the characteristic function $\chi _E$ of $E$. Show that $\chi _E$ is continuous iff $E$ is a closed and open subset of $X$.

Consider topological spaces $X_i=(X,\Omega_i),i=1,2$ such that $\Omega_1$ is finer than $\Omega_2$. Show that $f:X_1 \rightarrow Y$ is continuous if so is$f:X_2 \rightarrow Y$.

Let $X$ be a topological space and $Y$ be a topological space with ordered topology. Let $f,g:X \rightarrow Y$ be continuous functions. Show that the set $U:=\{x \in X:f(x)>g(x) \}$ is open in $X$.

Prove: Composition of continuous functions is continuous.

Let $X:=[0,1]$ and $A_0:=\{0\} , A_n:=[1/n,1]$. Then the function $f:[0,1] \rightarrow \mathbb{R}$given by $f(0)=1,f(x)=1/x(0<x\le1)$ is discontinuous at 0. However, $f|_{A_n}$ is continuous for every $n \ge 0$.

Show that the function $g: (0,1) \rightarrow \mathbb{R}$ given below is continuous on irrationals and discontinuous on rationals:

$g(x) = \frac{1}{q} \; if \; x \in \mathbb{Q} \; \cap \; (0,1)$and $x = \frac{p}{q}$ in reduced

form = 0 otherwise.

We say that a set is a $G_{\delta}$ set if it is countable intersection of open sets.

Show that a bijective continuous map from a compact metric space into a metric space sends closed sets to closed sets, and hence it is a homeomorphism.

Define $f: \mathbb{R} \rightarrow [0,1]$ by $f(x) =x$ if $|x| \le 1,$ and $f(x)= \frac{1}{|x|}$ if $|x| \ge1$. Then f is continuous on $\mathbb{R}$. Note that $f$ is onto but not one-one.

 Show that the interval $(a,b) \subseteq \mathbb{R}$ is homeomorphic to any other interval $(c,d) \subseteq \mathbb{R}$.

Show that $e^{-x}$ is a homeomorphism from $(0, \infty)$ onto $(0,1)$.