$f$ cannot be continuous on $[0,1]$
7358
For each $y \in \mathbb{R}$ either no $x$ with $f(x) = y$ or two such values of $x$. Show that $f$ is discontinuous.
21301
If nonnegative $f: [0,1] \rightarrow \mathbb{R}$ has a continuous $f''$, then $\int_0^1 \Big| \frac{f''(x)}{f(x)} \Big| \,dx >4$
101701
Prove that $\ln x$ is not uniformly continuous on $(0,1]$
5251
If $F$ is closed, does there exist a finite measure on $[0,1]$ with support $F$?
6186
An Integrable function which is not essentially bounded on any subinterval of $[0,1]$
148266
Let $f:X\to X $ be continuous. Does $f $ have a fixed point when $X=[0,1)$ or $X=(0,1) $?
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 $f(x) = f(x+\frac{1}{n})$.
47231
The space $C^0([0;1],\mathbb{R})$ of all continuous, real-valued functions on $[0,1]$ is not reflexive.
112501
A funtion and its fourier transformation cannot both be compactly supported unless f=0
76790
Condition to guarantee $f=0$ on $[a,b]$
129087
Proof that a sequence of continuous functions $(f_n)$ cannot converge pointwise to $1_\mathbb{Q}$ on $[0,1]$
15095
continuous onto map from $(0,1)\to (0,1]$
3518
Proving $f(x)=1/x$ on $(0,1 )$ is not uniformly continuous
50649
On the existence of a continuous bijection $f\colon [0,1]\to [0,1]\times [0,1]$
83338
Show that there's no continuous function that takes each of its values $f(x)$ exactly twice.
14578
Proving that an additive function $f$ is continuous if it is continuous at a single point
120406
$f$ strictly increasing does not imply $f'>0$