# Questions tagged [discontinuous-functions]

For questions about discontinuous functions, a function which for certain values or between certain values of the variable does not vary continuously as the variable increases.

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### Points of discontinuity and non differentiability of $| \sin(\pi/x)|$?

What are the points of discontinuity and non-differentiability of $| \sin(\pi/x)|$? I tried finding out the points of discontinuity for the function but couldn't understand why would a mod function ...
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### Example of a discontinous pseudo contraction mapping

$T: X\to X$ is a mapping with a fixed point $x^*$ with a property $\|T(x)-x^*\|\le \alpha\|x-x^*\|\forall x\in X,\alpha\in[0,1)$, could anyone give an example of such a map but discontinous? or ...
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### If $f$ is monotonic on $(a,b)$, the set of points of (a,b) at which $f$ is discontinuous is at most countable.

Now as an undergraduate student, I am studyign baby Rudin. I know the proof of this theorem are already well explained on match stack exchange here, but I have some question about the proof. In page ...
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### showing that a continuous function attains each of its values(or each real number in general) exactly 3 times.

The question and its answer is given below: But I am wondering how I can prove that the function $f$ attains each real number exactly 3 times, could anyone show me how can I do this? I ...
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### Why the function has removable discontinuity

$f(x) = \begin{cases} 2x-1 & \text{when }x<2 \\ 5 & \text{when }x=2 \\ \frac{1}{2}x + 2 & \text{when }x>2 \end{cases}$ I am learning Calculus But I can't seem to understand why this ...
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### Real valued function which is continuous only on transcendental numbers

First of all, I am sorry for asking this question. We know that $R$ is uncountable. And also the set of all transcendental numbers is uncountable. How can I construct a function $f(x)$ on $R$ which ...
### Is every measure $0$ set a set of discontinuities of a Riemann integrable function?
Let $f:[a,b]\rightarrow\mathbb{R}$ be bounded, and let $D$ be its set of discontinuities. Then Lebesgue's criterion states that $f$ is Riemann-integrable if and only if $D$ has Lebesgue measure $0$. ...
I am experimenting with the following theorem: A function $f:A\rightarrow B$ is continuous iff $f^{-1}(O)$ is open in A for every open set $O\subset B$. I am trying to find an open set in the ...