If $f$ is continuous then there exists $x\in [0,1]: f(x)=x$
30455
Show that any continuous $f:[0,1] \rightarrow [0,1]$ has a fixed point $\zeta$
90565
If $f:[0,1] \rightarrow \mathbb{R}$ is continuous function such that $f(0)=f(1)$ then there exists $ x\in [0,1]$ such that $f(x) = f(x+\frac{1}{n})$.
29154
$f$ is continuous and open implies $f$ injective
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).
36166
Uniform continuity on (0,1) implies boundedness
53627
Proving continuity of $f$
69869
if $f$ is continuous on $\mathbb{R}$ and $f(r)=0,r \in \mathbb{Q}$, then $f(x)=0,x \in \mathbb{R}$
104369
If $f$ continuous and $f(x^2) = f(x)$, then $f$ is a const
7876
If $f(x+y)=f(x)+f(y) ,\forall\;x,y\in\Bbb{R}$, then if $f$ is continuous at $0$, then it is continuous on $\Bbb{R}.$
29582
Continuous mapping $f: [0,1]\rightarrow (0,1)$ CSIR December $2013$
141945
Why is $f(x) = \sqrt{x}$ is continuous at $x=0$? Does $\lim_{x \to 0^{-}} \sqrt{x}$ exist?
61500
If f is continous and exist real numbers show that $f(x)=0$
145919
If $f: \Bbb{R}\rightarrow\Bbb{R}$ is continuous at $0$ and $f(x)=f(2x)$ for each $x\in\Bbb{R}$ then $f$ is constant.
55605
If $f_n(x_n) \to f(x)$ whenever $x_n \to x$, show that $f$ is continuous
37966
If $f$ is continuous and $f(x+y) = f(x)+f(y)$, then $f(x) = cx$ for all $x \in \mathbb{R}$
131992
A problem on continuous functions
139744
For a continuous function $f$ show $\exists c\in (0,1)$ s.t $f(c)=3c^2$