Prove that there does not exist a continuous function over reals for which it takes on every value 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.
33330
Continuous Functions from $\mathbb{R}$ to $\mathbb{Q}$
2142
Finding a set of continuous functions with a certain property 2
100436
Examples of nowhere continuous functions
135449
Prove: $\frac1x$ is not a uniformly continuous function
20091
Every continuous function is the difference of two nonnegative continuous functions
13257
How to prove this problem about uniform continuity
98356
There does not exist an entire function which satisfies $f({1\over n})={1\over 2^n}$?
58816
Existence of a continuous function which does not achieve a maximum.
12240
Problem with a continuous function
120935
Why is every continuous function on the reals determined by its value on rationals?
102075
A function continuous on all irrational points
87027
There is no continuous injective function $f:\mathbb{R}^n\longrightarrow\mathbb{R}^m$
57014
Existence of a power series converging non-uniformly to a continuous function
158309
Find a function that is continuous only on rationals
50111
non-continuous function satisfies $f(x+y)=f(x)+f(y)$
139763
A function takes every function value twice - proof it is not continuous