Prove that if f is continuous in A then |f| is also continuous.
44574
Continuity of absolute value of a function
74364
Show that function $f$ has a continuous extension to $[a,b]$ iff $f$ is uniformly continuous on $(a,b)$
69716
$f_x$ is Borel measurable and $f^y$ is continuous then $f$ is Borel measurable
129073
If $f'$ is bounded, then $f$ is uniformly continuous
69869
if $f$ is continuous on $\mathbb{R}$ and $f(r)=0,r \in \mathbb{Q}$, then $f(x)=0,x \in \mathbb{R}$
14578
Proving that an additive function $f$ is continuous if it is continuous at a single point
30965
$f$ is continuous on $X$ iff $f$ is continuous on every compact subset
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 $\forall\alpha\in\mathbb{R}$
72694
Proving that $f(x) = g(x)$ for some $x \in [a,b]$ if $f,g$ continuous, $f(a)  g(b)$
100740
proof of that a continuous function has a fixed point
95970
Let $f:[0,1] \rightarrow \mathbb{R}$ be a continuous function. If $f(0)=f(1)$ then exists $x\in [0,1]$ s.t. $f(x) = f(x+\frac{1}{2})$
153730
Show that $f(x)=x^2$ is continuous at $a=2$ using the $\delta-\epsilon$ definition of continuity.
57642
Uniform Continuity implies Continuity
141223
$f(x^2) = 2f(x)$ and $f(x)$ continuous
144614
If $f$ is continuous on $[a,b]$ and $F(x) = \sup f([a,x])$. Prove that $F$ is continuous on $[a,b]$ .
130299
Prove that if a function $f: X\to Y$ continuous then its graph is closed
58664
Proving a function is continuous for a fixed variable