Show that a continuous function doesn't exist s.t. for every real $c$, there exist $exactly$ two reals $a, b$ s.t. $f(a)=f(b)=c$.
100001
Show that a continuous function doesn't exist such that for every real $c$, there exist exactly two reals $a, b$ s.t. $f(a)=f(b)=c$.
58816
Existence of a continuous function which does not achieve a maximum.
145017
Example of a function $f$ which is nowhere continuous but $|f|$ should be continuous at all points
129716
Let $f$ be continuous on $ [a,b]$, differentiable on $ (a,b) $ and $ f(x) \neq 0 \forall x \in (a,b) $ Prove $ \exists c \in (a,b) $ such that
3922
How to prove that there exist no functions $f,g:\Bbb{R}\to\Bbb{R}$ such that $f(g(x))=x^{2018}$ and $g(f(x))=x^{2019}$?
20091
Every continuous function is the difference of two nonnegative continuous functions
113687
$f$ is continuous function from $ [0,1]$ to itself and f(0)=0, f(1)=1 and $f(f(x))=x$ then show that $f(x)=x$ on$[0,1]$
99906
If $f'(c)$ exists then $\lim_{x\to c} f(x)=f(c)$
133682
Prove that if $f(t)$ satisfies $f(t)=f'(t)$, then $f(t+s)=f(t)f(s)$ for all $t,s\in R$ without using the fact that $f(t)=c\cdot e^t$.
61640
Prove c exists for f:[0,1] -> R
234
No continuous function switches $\mathbb{Q}$ and the irrationals
143354
Prove that there exists $c\in\mathbb{R}$ such that $f'(c)=f(c)$
18222
Set of continuity points of a real function
151427
f, g continuous for all rationals follow by continuous on all reals?
37966
If $f$ is continuous and $f(x+y) = f(x)+f(y)$, then $f(x) = cx$ for all $x \in \mathbb{R}$
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?
125682
Proof verification: the product of two continuous functions is continuous