Let $(X,d_X)$ and $(Y, d_Y)$ be two metric spaces. If $f: X\to Y$ is continuous, prove that $$ f(\overline{E})\subset \overline{f(E)}\text{ for any }E\subset X. $$
Let $A$ and $B$ be disjoint nonempty closed sets in a metric spaces $X$ and define $$ f(x)=\frac{\rho_A(x)}{\rho_A(x) +\rho_B(x)},\: x\in X. $$ Show that $f$ is a continuous function on $X$ whose range lies in $[0,1]$, $f(x)=0$ for $x\in A$ and $f(x)=1$ for $x\in B$.
Suppose that $(f_n)_{n\in\mathbb{N}}$ is a sequence of uniformly continuous functions on $\mathbb{R}$, which converges uniformly to a function $g:\mathbb{R}\to\mathbb{R}$. Prove or disprove: $g$ is also uniformly continuous.
Let $f:\mathbb{R}\to \mathbb{R}$ be given. Prove or disprove:
(a) If $f$ is uniformly continuous, then so is $f^2$.
(b) If $f$ is uniformly continuous, then so is $\frac{f^2}{1+f^2}$.
Let $f\in \operatorname{C}(\mathbb{R}^n,\mathbb{R})$ be a continuous function, and $\alpha$ be a real number. Prove by giving a complete argument or disprove by exhibiting a counterexample that $$ \partial\big\{x\in \mathbb{R}^n\, \big |\, f(x)>\alpha\big\}=\big\{ x\in\mathbb{R}^n\, \big |\, f(x)=\alpha\big\}. $$
Consider the sequence $(f_n)_{n\in\mathbb{N}}$ defined through $$ f_n(x)=\frac{1}{1+n^2 x^2}\, ,\: x\in \mathbb{R}. $$ For what values of $x$ does the series $ \sum_{n=1}^\infty f_n$ converge absolutely? On what intervals does it converge uniformly ? On what interval does it fail to converge uniformly? Is $f$ continuous wherever the series converges? Is $f$ bounded?
Let the $I$ be given by $$I(x)=\begin{cases} 0\, ,&\text{if }x\le 0\, ,\\ 1\, ,&\text{if }x>0\, . \end{cases} $$ Let $(x_n)_{n\in \mathbb{N}}$ be a sequence of distinct points of $(a,b)$. Assume that $\sum_{n=1}^\infty |c_n| $ converges and prove that the series $$ f(x)=\sum_{n=1}^\infty c_n I(x-x_n)\, ,\: a\le x\le b\, , $$ converges uniformly on $[a,b]$. Show that $f$ is continuous for every $x\ne x_n$.
Suppose that $(f_n)_{n\in\mathbb{N}}$ and $(g_n)_{n\in\mathbb{N}}$ are
two sequences of functions defined on a set $E$ satisfying
(i) $\sum_{n=1}^\infty f_n$ has uniformly bounded partial sums.
(ii) $g_n\to 0$ uniformly on $E$.
(iii) $g_1\ge g_2\ge g_3\ge \dots $.
Prove that $\sum_{n=1}^\infty f_n g_n$ converges uniformly on $E$.
Let $x\in \mathbb{R}$ and $n\in \mathbb{N}$ and define $$ f_n(x)=\frac{e^{nx}}{e^{nx}+e^{-nx}}. $$ Show that $(f_n)_{n\in\mathbb{N}}$ converges pointwise on $\mathbb{R}$ and find its limit. Is the convergence uniform on $[0,1]$? Is it on $[1,3]$?
Let $f:\overline{\mathbb{B}}(0,1)\to(0,\infty)$. Determine the convergence properties of the infinite series $$ \sum _{n\in \mathbb{N}}e^{-nf(x)}. $$ Does it converge pointwise? Uniformly? What is the limiting function?