Showing that $\ker T$ is closed if and only if $T$ is continuous.
108734
$T$ is continuous if and only if $\ker T$ is closed
118207
Prove that the product of two continuous functions is continuous
31266
Proof of "the continuous image of a connected set is connected"
147247
Continuous function on closed unit ball
113503
Show that exists a not decreasing function that $f:(a,b)\rightarrow \mathbb{R}$ that is continuous only in $(a,b)\setminus D$.
91682
Graph of a continuous function is closed
72240
Prove that if a linear operator is continuous, then it is bounded.
113975
Let $f:\mathbb{R}\to\mathbb{R}$, $f(kx)=f(x)$. Show that if $f$ is continuous at $0$ then $f$ is a Constant function
40209
$f : S^1 \to\mathbb R$ is continuous then $f(x)=f(-x)$ for some $x\in S^1$
153495
Continuous function does not map closed set to closed set
74950
Show a function for which $f(x + y) = f(x) + f(y) $ is continuous at zero if and only if it is continuous on $\mathbb R$
58664
Proving a function is continuous for a fixed variable
16735
If $f,g:X \to Y$ are continuous and $Y$ is $T_2$, then $\{x \in X\,|\,f(x)=g(x)\}$ is closed
33129
A linear map $S:Y^*\to X^*$ is weak$^*$ continuous if and only if $S=T^*$ for som $T\in B(X,Y)$
151573
Show f is continuous on (0, ∞) iff f is continuous at 1 for a defined function.
8414
Proof verification: Set of partial limits of a sequence is closed
3649
Closed unit interval is connected proof