If $f:[0,1]\to\Bbb R$ is continuous and $f(0)=f(1)$ show that $\forall n\in\Bbb N,\exists x_n,y_n\in[0,1]:|x_n-y_n|=1/n$ and $f(x_n)=f(y_n)$
12632
Universal Chord Theorem
3518
Proving $f(x)=1/x$ on $(0,1 )$ is not uniformly continuous
157821
Let $S$ be a finite set. Show that any function $f: S \rightarrow \mathbb{R}^n$ is continuous on S. What if $S$ is countably infinite?
29582
Continuous mapping $f: [0,1]\rightarrow (0,1)$ CSIR December $2013$
74950
Show a function for which $f(x + y) = f(x) + f(y) $ is continuous at zero if and only if it is continuous on $\mathbb R$
111485
Let $f(x)$ be continuous on [$0,1$] to $f(x)=0$ for each $x\in[0,1]$
108115
If $f: [0,1]\rightarrow \mathbb{R}$ is continuous function positive
149354
Proof of continuity $f(x+y) = f(x) + f(y)$
22061
$|f(x)-f(y)|\geq k|x-y|$.Then $f$ is bijective and its inverse is continuous.
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 $(0, 1)$?
122160
$f:\mathbb R \to \mathbb R$ be a continuous such that $f(f(f(x)))=x,\forall x \in \mathbb R$ , is $f(x)=x , \forall x \in \mathbb R$?
154559
For each $x\in X$ there exists a neighborhood of x s.t $A\cap U\neq \emptyset$ for only finitely many $A\in\mathcal{A}$. Show that f is continuous.
82405
For f continuous on $[0,1]$, show that there exist points $\alpha_k$ such that $\sum \limits_{k=1}^n \frac{1}{f'(\alpha_k)} = n $
143174
property of continuous function
44144
If $f:[a,b]\to \mathbb R$ is continuous then so is $\max_{t\in [a,x]}f(t)$?
57023
If $f,g$ are continuous at $a$, show that $h(x)=\max\{f(x),g(x)\}$ and $k(x)=\min\{f(x),g(x)\}$ are also continuous at $a$