Show that there's no continuous function that takes each of its values $f(x)$ exactly twice.
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.
136538
How can I show that a bijection between $[0,1)$ and $(0,1)$ cannot be continuous?
2321
Can there be two distinct, continuous functions that are equal at all rationals?
19146
Showing that $f$ has exactly one fixed point
130019
Function that is continuous only on the transcendental numbers
95970
Let $f:[0,1] \rightarrow \mathbb{R}$ be a continuous function. If $f(0)=f(1)$ then exists $x\in [0,1]$ s.t. $f(x) = f(x+\frac{1}{2})$
18082
A continuous function that attains neither its minimum nor its maximum at any open interval is monotone
6186
An Integrable function which is not essentially bounded on any subinterval of $[0,1]$
110458
Show that if $f$ is continuous on $(0,1)$ and $\lim_{x \to 0}f(x) = + \infty$ then $f$ is not uniformly continuous
27800
For continuous function $ f:\mathbb S^1 \to \mathbb R$ there exists uncountably many distinct points $x,y$ such that $f(x)=f(y)$
13852
Assume $f$ is a continuous one-to-one function over an interval. Prove that $f$ is strictly monotone
97619
Which function ($f$) is continuous nowhere but $|f(x)|$ is continuous everywhere?
14578
Proving that an additive function $f$ is continuous if it is continuous at a single point
97104
Proof that continuous function on $ [0, 2]$ has $f(x) = f(x-1)$ for some $ x$
118048
Can a function with just one point in its domain be continuous?
119060
What can you say about a continuous function that is zero at all integer values?
76009
Can a continuous function from the reals to the reals assume each value an even number of times?