Universal rope if for every continuous function $f:[0,1] \rightarrow \mathbb{R}$ with $f(0)=f(1)$
12632
Universal Chord Theorem
108115
If $f: [0,1]\rightarrow \mathbb{R}$ is continuous function positive
91947
Proof of $ f:I_n \rightarrow I_m \Rightarrow n = m$
132013
How do I show that if $f : [0,1] \to [0,1]$ is continuous, then there is some $c$ such that $f(c) = c$?
15487
Continuous function from a connected set?
13852
Assume $f$ is a continuous one-to-one function over an interval. Prove that $f$ is strictly monotone
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 $
117502
Proving that a certain continuous function is surjective.
107727
$m\{x\in [0,1]:f'(x)=0\}>0$
148813
Prove: $f: \mathbb{R} \rightarrow \mathbb{R}$ st for every $x \in \mathbb{R}$ there exists $n$ st $f^{(n)}(x) = 0$, f is a polynomial.
108997
Continuous functions that attain local extrema at every point
143798
Prove that, there exists no continuous function $f:\mathbb R\rightarrow\mathbb R$ with $f=\chi_{[0,1]}$ almost everywhere.
102075
A function continuous on all irrational points
131992
A problem on continuous functions
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 $x$
114856
Universal property of initial topology
88586
show that every continuous real-valued function defined on $S_{\mathbb{\Omega}}$ is eventually constant