Assume $f : [0, 1] \rightarrow [0, 1]$ is continuous. Show that there must be a point $x \in [0, 1]$ such that $f(x) = x$
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proof of that a continuous function has a fixed point
97104
Proof that continuous function on $ [0, 2]$ has $f(x) = f(x-1)$ for some $ x$
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 $x$
30455
Show that any continuous $f:[0,1] \rightarrow [0,1]$ has a fixed point $\zeta$
36166
Uniform continuity on (0,1) implies boundedness
129324
Let $f:[0;1]\to\mathbb R$ be continiously differentiable
104369
If $f$ continuous and $f(x^2) = f(x)$, then $f$ is a const
13765
Is a rational-valued continuous function $f\colon[0,1]\to\mathbb{R}$ constant?
136011
Must $f$ be continuous?
154521
Linear functional $f$ is continuous at $x_0=0$ if and only if $f$ is continuous $\forall x\in X$?
148266
Let $f:X\to X $ be continuous. Does $f $ have a fixed point when $X=[0,1)$ or $X=(0,1) $?
92089
If $f:[a,b]\to \mathbb{R}$ is continuous and nonnegative and $\int_a^b{f}=0$, then $f(x)=0$ for all $x\in [a,b]$
111756
How do I prove that then $f(x) = 1$ for some $x ∈ [0, 1]$?
13927
Values of $a$ s.t. for all continuous $f$ with $f(0)=f(1)$ there exists $x$ s.t. $f(x+a) = f(x)$
129087
Proof that a sequence of continuous functions $(f_n)$ cannot converge pointwise to $1_\mathbb{Q}$ on $[0,1]$
42853
If $f \geq 0$ is continuous and $\int_{a}^{b} f(x) \, dx = 0$, then $f =0$
151573
Show f is continuous on (0, ∞) iff f is continuous at 1 for a defined function.