Prove that $f(x)=\begin{cases}x+1&\text{if}&x\in(-\infty,-1)\\x-1 &\text{if}&x\in(1, +\infty)\\0&\text{if}&x=0\end{cases}$ is continuous
118048
Can a function with just one point in its domain be continuous?
80643
Show that f(x) defined on [0,infinity) R to is continuous at 0.
55605
If $f_n(x_n) \to f(x)$ whenever $x_n \to x$, show that $f$ is continuous
158888
Prove that the function $f(x)=\frac{x}{x^2+1}$ is continuous at $x=-1$.
94781
Prove that $M(t)=\sup_ {a \leq x \leq t} f(x)$ given $f(x)$ is continuous on $[a,b]$
58664
Proving a function is continuous for a fixed variable
72694
Proving that $f(x) = g(x)$ for some $x \in [a,b]$ if $f,g$ continuous, $f(a)  g(b)$
7876
If $f(x+y)=f(x)+f(y) ,\forall\;x,y\in\Bbb{R}$, then if $f$ is continuous at $0$, then it is continuous on $\Bbb{R}.$
37966
If $f$ is continuous and $f(x+y) = f(x)+f(y)$, then $f(x) = cx$ for all $x \in \mathbb{R}$
149354
Proof of continuity $f(x+y) = f(x) + f(y)$
69869
if $f$ is continuous on $\mathbb{R}$ and $f(r)=0,r \in \mathbb{Q}$, then $f(x)=0,x \in \mathbb{R}$
129716
Let $f$ be continuous on $ [a,b]$, differentiable on $ (a,b) $ and $ f(x) \neq 0 \forall x \in (a,b) $ Prove $ \exists c \in (a,b) $ such that
122160
$f:\mathbb R \to \mathbb R$ be a continuous such that $f(f(f(x)))=x,\forall x \in \mathbb R$ , is $f(x)=x , \forall x \in \mathbb R$?
118332
proof $f(\overline{A})\subseteq \overline{f(A)} \Leftrightarrow f$ continuous
100740
proof of that a continuous function has a fixed point
134752
Proof (epsilon delta) for the continuity of a function at a point
72044
prove $x \mapsto x^2$ is continuous