$f:S^1 \rightarrow R$ continuous means there exists a point $x$ such that $f(x) = f(-x)$.
40209
$f : S^1 \to\mathbb R$ is continuous then $f(x)=f(-x)$ for some $x\in S^1$
141945
Why is $f(x) = \sqrt{x}$ is continuous at $x=0$? Does $\lim_{x \to 0^{-}} \sqrt{x}$ exist?
118332
proof $f(\overline{A})\subseteq \overline{f(A)} \Leftrightarrow f$ continuous
18222
Set of continuity points of a real function
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 $\Bbb{R}.$
17234
Show that a continuous function has a fixed point
53627
Proving continuity of $f$
151573
Show f is continuous on (0, ∞) iff f is continuous at 1 for a defined function.
127114
Differentiability implying continuity
152784
$f \in {\mathscr R[a,b]} \implies f $ has infinitely many points of continuity.
154572
Continuity of a function at an isolated point
2093
If $f$ is continuous then $G$ is homeomorphic to $X$.
39406
Continuity $\Rightarrow$ Intermediate Value Property. Why is the opposite not true?
57642
Uniform Continuity implies Continuity
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 $\forall\alpha\in\mathbb{R}$
104044
Prove continuous functions $f$ and $g$ are equal
124444
If $f$ s discontinuous at $x_0$ but $g$ is continuous there, then $f+g$?