$f$ is additive and continuous at a point $\implies$ $f(x) =ax$, for some $a \in \mathbb R$
13086
Overview of basic facts about Cauchy functional equation
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$f(x^2) = 2f(x)$ and $f(x)$ continuous
8836
If $f$ is continuous on $[a , b]$ and $\int_a^b f(x) p(x)dx = 0$ then $f = 0$
5373
Additive function $T: \mathbb{R} \rightarrow \mathbb{R}$ that is not linear.
44144
If $f:[a,b]\to \mathbb R$ is continuous then so is $\max_{t\in [a,x]}f(t)$?
127868
If $f$ is nonnegative and integrable then $F(x) = \int_{-\infty}^x f$ is continuous.
105652
Suppose $f: \mathbb{R} \to \mathbb{R}$ is function such that $f^3$ is continuous on $\mathbb{R}$. Prove that $f$ is continuous on $\mathbb{R}$
104369
If $f$ continuous and $f(x^2) = f(x)$, then $f$ is a const
71049
Functional equation $f(xy)=f(x)+f(y)$ and continuity
90565
If $f:[0,1] \rightarrow \mathbb{R}$ is continuous function such that $f(0)=f(1)$ then there exists $ x\in [0,1]$ such that $f(x) = f(x+\frac{1}{n})$.
132013
How do I show that if $f : [0,1] \to [0,1]$ is continuous, then there is some $c$ such that $f(c) = c$?
69695
$f\geq 0$, continuous and $\int_a^b f=0$ implies $f=0$ everywhere on $[a,b]$
72006
Suppose $f(x)$ is linear i.e. $f(x + y) = f(x) +f(y) $ and monotone on $[-\infty, +\infty]$, then $f(x) = ax$, a real.
153730
Show that $f(x)=x^2$ is continuous at $a=2$ using the $\delta-\epsilon$ definition of continuity.
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Let $f(x)$ be continuous on [$0,1$] to $f(x)=0$ for each $x\in[0,1]$
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If $f$ continuous and $\lim_{x\to-\infty }f(x)=\lim_{x\to\infty }f(x)=+\infty $ then $f$ takes its minimum.
97104
Proof that continuous function on $ [0, 2]$ has $f(x) = f(x-1)$ for some $ x$