Function on $[0,1]$ which takes every value twice has infinitely many discontinuities.
3276
Function $f:[0,1] \to [0,1]$ taking on each value in $[0,1]$ exactly twice
52486
Is there a function that is contiunous at all the rationals, and discontinuous at all the irrationals?
152563
Set of discontinuity of monotone function is countable
90685
The composition of non-linear entire functions with no fixed points has infinitely many fixed points.
64836
Functions that are always less than their derivatives
44057
Pointwise but not uniform convergence of continuous functions on $[0,1]$
24817
Prove continuity on a function at every irrational point and discontinuity at every rational point.
15820
Discontinuous linear functional
74793
The primitive of a discontinuous function?
97104
Proof that continuous function on $ [0, 2]$ has $f(x) = f(x-1)$ for some $ x$
108997
Continuous functions that attain local extrema at every point
150833
Discontinuities of the derivative of a differentiable function on closed interval
131992
A problem on continuous functions
98356
There does not exist an entire function which satisfies $f({1\over n})={1\over 2^n}$?
17642
Why are removable discontinuities even discontinuities at all?
21269
Bijective continuous function between $(0 ,\infty)$ and $\mathbb{R}$.
148910
Multivariate polynomials have infinitely many zeroes