$f$ is continuous iff $G(f)$ is a closed set in metric spaces
130299
Prove that if a function $f: X\to Y$ continuous then its graph is closed
40915
Proof if $f$ continuous in $x_0$ then there is a neighbourhood of $x_0$ so f bounded.
124444
If $f$ s discontinuous at $x_0$ but $g$ is continuous there, then $f+g$?
112303
Distances between closed sets on metric spaces
84512
The preimage of continuous function on a closed set is closed.
75479
Proving that this function is continuous on $G\times G$
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 $a$
12523
Supremum of a Continuous Function is Continuous
88185
Proof that a subspace $A$ of a complete metric space $X$ is complete iff $A$ is closed
83204
Is every $G_\delta$ set the set of continuity points of some function $f$?
6415
Distance function is continuous in topology induced by the metric
29154
$f$ is continuous and open implies $f$ injective
2093
If $f$ is continuous then $G$ is homeomorphic to $X$.
58597
A characterization of functions from $\mathbb R^n$ to $\mathbb R^m$ which are continuous
42638
Showing that a function $f$ has a unique fixed point in a metric space.
151573
Show f is continuous on (0, ∞) iff f is continuous at 1 for a defined function.
93021
$f:X\longrightarrow\mathbb{R}$ is continuous iff $\{x\in X:f(x)\geq\alpha\}$ and $\{x\in X:f(x)\leq\alpha\}$ are closed $\forall\alpha\in\mathbb{R}$