Let $f: [0, 1]\rightarrow R$ be continuous and suppose that $f(0) = f(1)$. Prove there is $x$ such that $f(x)=f(x+\frac{1}{n})$
12632
Universal Chord Theorem
104044
Prove continuous functions $f$ and $g$ are equal
61500
If f is continous and exist real numbers show that $f(x)=0$
111756
How do I prove that then $f(x) = 1$ for some $x ∈ [0, 1]$?
112512
How to prove that there are finitely many $n$ such that $f(1+1/n) = \frac{1}{n+1}$?
158888
Prove that the function $f(x)=\frac{x}{x^2+1}$ is continuous at $x=-1$.
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$[0,1]$
121771
Proof $f(x)\equiv 0$
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?
30860
Prove that $f(x)=0$ in $[-1,1]$.
139744
For a continuous function $f$ show $\exists c\in (0,1)$ s.t $f(c)=3c^2$
132013
How do I show that if $f : [0,1] \to [0,1]$ is continuous, then there is some $c$ such that $f(c) = c$?
95469
Suppose $f:[0,1] \Rightarrow \mathbb{R}$ is continuous and $\int_0^x f(x)dx = \int_x^1 f(x)dx$. Prove that $f(x) = 0$ for all $x$
111990
Continuous function from $(0, 1]$ onto $(0, 1)$?
54403
Given $f : \Bbb{R} \to \Bbb{R}$ is a continuous function. To prove that $\int_{0}^{1}f(x)x^2 dx = \frac{f(c)}{3}$ for some $c \in [0,1]$
112636
How many continuous function $f(x)$ exist such that $\int_{0}^{1}f(x)\big(1-f(x)\big)\mathrm dx = \frac{1}{4}$?
63889
Prove that if $u_n\rightarrow \infty$ then $\frac{1}{u_n} \rightarrow 0$