Show that every mapping $f: X \to Y$ is continuous where $(X,d)$ is a metric space with the discrete metric
123972
Proving that every function defined on a discrete metric space is uniformly continuous.
97914
Contractive mapping on compact space
101319
Properties of a continuous map $f : \mathbb{R}^2\rightarrow \mathbb{R}$ with only finitely many zeroes
158349
Prove the map sending a matrix to it's inverse is continuous.
140052
Show that $f(X - D) \cap f(D) = \varnothing$ with $f$ continuous in $X$, $D$ dense in $X$ and $f|_{D}$ homeomorphism
100740
proof of that a continuous function has a fixed point
55238
Show that if $f:X\to Y$ is a continuous function if and only if the graph of $f$ is a closed subset of $X\times Y$
58597
A characterization of functions from $\mathbb R^n$ to $\mathbb R^m$ which are continuous
128894
About compactness and completeness of set $X = \mathbb{R}$ with the metric $d(x, y) = \frac{|x-y|}{1+|x-y|}$
130537
Proving that $d(f,g)=\|f-g\| = \sup \limits_{0\leq x \leq 1} |f(x)-g(x)|$ is a metric on $X=C[0,1]$
153952
A map between metric spaces preserving convergent sequences is continuous
57023
If $f,g$ are continuous at $a$, show that $h(x)=\max\{f(x),g(x)\}$ and $k(x)=\min\{f(x),g(x)\}$ are also continuous at $a$
64619
Continuity of a function of product spaces
72044
prove $x \mapsto x^2$ is continuous
69777
Show that, for every metric $d$, the metrics $d/(1+d)$ and $\min\{1,d\}$ are equivalent
34344
Is a function that maps every compact set to a compact set continuous?
42266
Show that $d(x,y)=\min \{1,|x-y|\}$ is a metric on $\mathbb R$