$g(x+y)=g(x)g(y)$ for all $x,y \in\mathbb{R}$. If $g$ is continuous at $0$, prove that $g$ is continuous on $\mathbb{R}$
62723
If $f\colon \mathbb{R} \to \mathbb{R}$ is such that $f (x + y) = f (x) f (y)$ and continuous at $0$, then continuous everywhere
149399
Given continuous $f$ and $g$, there exists $c \in (a,b)$ such that $\int_{a}^{c} f(x) \ \mathrm dx = \int_{c}^{b} g(x) \ \mathrm dx$
94781
Prove that $M(t)=\sup_ {a \leq x \leq t} f(x)$ given $f(x)$ is continuous on $[a,b]$
103932
How can I prove that a polynomial with degree $n$ is continuous everywhere in $\mathbb{R}$ using definitions?
144614
If $f$ is continuous on $[a,b]$ and $F(x) = \sup f([a,x])$. Prove that $F$ is continuous on $[a,b]$ .
67234
The function $G: x \mapsto 2^{x^2}$ maps $\mathbb{R}$ onto $\{ x \in \mathbb{R} : x \geq 1 \}$
143174
property of continuous function
133422
How to prove the continuity of an induced map; $g_* : C(X, Y) \rightarrow C(X, R)$, induced from the map $g \in C(Y, R)$, endowed with fine topology.
71049
Functional equation $f(xy)=f(x)+f(y)$ and continuity
50111
non-continuous function satisfies $f(x+y)=f(x)+f(y)$
40609
Prove that $f(X)$ and $g(Y)$ are independent if $X$ and $Y$ are independent
124444
If $f$ s discontinuous at $x_0$ but $g$ is continuous there, then $f+g$?
138105
Let $f$ and $g$ be continuous at $ x=a$, with $f(a)<g(a)$. Prove that there exists a neighborhood $I$ of a such that $f(x)<g(x)$ for all $x \in I$.
122160
$f:\mathbb R \to \mathbb R$ be a continuous such that $f(f(f(x)))=x,\forall x \in \mathbb R$ , is $f(x)=x , \forall x \in \mathbb R$?
116732
Proving that a continuous map is surjective
42084
Set of points of continuity are $G_{\delta}$
22303
How do I prove that $f(x)f(y)=f(x+y)$ implies that $f(x)=e^{cx}$, assuming f is continuous and not zero?