Proving that continuous function $f:[0,2] \to \mathbb{R}$ such as $f(0)=f(2)$, $\exists x \in [0,1]:f(x)=f(x+1)$
97104
Proof that continuous function on $ [0, 2]$ has $f(x) = f(x-1)$ for some $ x$
76148
Show that $\exists c \in \mathbb{R} : f(c) = 0$
14578
Proving that an additive function $f$ is continuous if it is continuous at a single point
131992
A problem on continuous functions
102143
$f:\mathbb R\rightarrow \mathbb R$ be a continuous function such that $\int_{0}^{\infty}f(x)dx$ exists.
46629
Prove if $f(x) = g(x)$ for each rational number x and $f$ and $g$ are continuous, then $f = g$
109942
Find continuous functions such that $f(x+y)+f(x-y)=2[f(x)+f(y)]$
35889
Suppose that f(0) = f(1) = 0 and x ∈ (0, 1). Prove that there exists c ∈ (0, 1) such that f'(c) = f(x).
109896
Proving that the function $f(x,y)=\frac{x^2y}{x^2 + y^2}$ with $f(0,0)=0$ is continuous at $(0,0)$.
54797
Find the continuous function defined over $[0,1]$ such that $\int_0^1f(x){\rm d}x=1, \int_0^1 xf(x){\rm d}x=a$ and $\int_0^1x^2f(x){\rm d}x=a^2 .$
124535
continuous function from $[\pi,2\pi]\to \mathbb{R}^2$
111756
How do I prove that then $f(x) = 1$ for some $x ∈ [0, 1]$?
21269
Bijective continuous function between $(0 ,\infty)$ and $\mathbb{R}$.
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?
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
104369
If $f$ continuous and $f(x^2) = f(x)$, then $f$ is a const
108115
If $f: [0,1]\rightarrow \mathbb{R}$ is continuous function positive