If $f=g$ a.e. and $f$ and $g$ are continuous, then $f=g$ every where.
72847
Does $f(x)$ is continuous and $f = 0$ a.e. imply $f=0$ everywhere?
153730
Show that $f(x)=x^2$ is continuous at $a=2$ using the $\delta-\epsilon$ definition of continuity.
104757
If $f \circ g$ is surjective, then f is surjective
40609
Prove that $f(X)$ and $g(Y)$ are independent if $X$ and $Y$ are independent
144614
If $f$ is continuous on $[a,b]$ and $F(x) = \sup f([a,x])$. Prove that $F$ is continuous on $[a,b]$ .
55368
$f$ is continuous, if $f_n$ continuous and $f_n\to f$ uniformly
44402
If $\{x:f(x) > a\}$ and $\{x:f(x) < a\}$ are open$\forall~a\in\mathbb{R}$ then f is continuous.
29154
$f$ is continuous and open implies $f$ injective
37966
If $f$ is continuous and $f(x+y) = f(x)+f(y)$, then $f(x) = cx$ for all $x \in \mathbb{R}$
143174
property of continuous function
120960
$f,g$ such that $\int fg = \int f \int g$
45222
When $f(x) = g(y)$ for almost every $(x,y)$, must $f$ and $g$ be constant almost everywhere?
86327
If the absolute value of a function is continuous, is the function continuous?
106060
If $f$ is continuous and with a limit at infinity then $f$ is uniformly continuous
20091
Every continuous function is the difference of two nonnegative continuous functions
117502
Proving that a certain continuous function is surjective.
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 $\forall\alpha\in\mathbb{R}$