If $f(x_1)=f(x_2)=y$, then prove that $f$ can not be continuous everywhere
76009
Can a continuous function from the reals to the reals assume each value an even number of times?
104044
Prove continuous functions $f$ and $g$ are equal
69122
If $f,g$ are uniformly continuous prove $f+g$ is uniformly continuous but $fg$ and $\dfrac{f}{g}$ are not
11158
Proof that a continuous function with $f(x) = f(x^2)$ is constant.
145017
Example of a function $f$ which is nowhere continuous but $|f|$ should be continuous at all points
102457
for any $A\subseteq X$, $f(\overline{A})\subseteq\overline{f(A)}$ , if and only if $f: X \to Y$ is continuous.
145368
How to show $f:Y\to X^{+}$ is continuous?
62817
Functions which are Continuous, but not Bicontinuous
77621
Continuity of function mapping connected set to connected set
55605
If $f_n(x_n) \to f(x)$ whenever $x_n \to x$, show that $f$ is continuous
51156
Function is Continuous When Restricted to Closed Sets Which Cover the Space
53627
Proving continuity of $f$
144001
If $f$ is continuous and $\,f\big(\frac{1}2(x+y)\big) \le \frac{1}{2}\big(\,f(x)+f(y)\big)$, then $f$ is convex
130023
Prove that the sum of two continuous functions is continuous
151573
Show f is continuous on (0, ∞) iff f is continuous at 1 for a defined function.
83326
$f:\mathbb R\to\mathbb R$ continuous function. Which of the following sets can not be image of $(0,1]$ under $f$?
20091
Every continuous function is the difference of two nonnegative continuous functions