# Questions tagged [discontinuous-functions]

Discontinuous functions in $\mathbb R$ are characterized as being "broken" when pictorially represented in graph form. More generally, a function $f:X\to Y$ is discontinuous at $x\in X$ if there exists an open set $V\subset Y$ such that $f(x)\in V$ and $x\in\operatorname{Bd}(f^{-1}(V))$. Use this tag to ask questions about discontinuous functions on the reals or on other topologies.

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### prove non-continuity using open sets (topology)

In topology, continuity is defined as: A function $f:X\rightarrow Y$ is continuous if the inverse image of an open set in $Y$ is an open set in $X$. I have a problem to use it to check the non-...
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### Is the following function $f$ continuous? [closed]

Say we have a function $f(x)$ = $\frac{1}{x}\\$ for all x $\in$ $\mathbb{R}$ such that x $\ne$ 0 and $f(0)$ = 2. Do we consider this function f continuous?
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### What is the intuition behind how discontinuous a derivative can be?

I’ve looked at some answers on this site about how discontinuous a derivative can be, and it seems there are some properties that a derivative must satisfy. Darboux’s theorem tells us that if we have ...
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### How does $(t^3, t^2)$ represent $x^{2/3}$?

I just learnt about this parameterization and somehow I am not being able to wrap my head around it. How on earth is the curve $\left<x, x^{2/3}\right> = \left<t^3, t^2\right>$. I got ...
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### Constructing $\sqrt{\sqrt{1+x^2}-1}$ to be smooth (cancelling the square)

$\DeclareMathOperator{\sign}{sign}$ Is there a way to rewrite $f(x)=\sign(x)\sqrt{\sqrt{1+x^2}-1}$ using (smooth) elementary functions? As far as I can see the function seems infinitely ...
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### How to prove that the n-th root is not continuous using the Fundamental Group

I need to prove that the function $g:\mathbb{C}^∗ \rightarrow \mathbb{C}^*$ such that $(g(z))^n=z$ is not continuous using the fundamental group. I tried to use the argument in the question: How to ...
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### How to prove that the complex logarithm is not continuous using the Fundamental Group

I need to prove that the function $f:\mathbb{C^*} \rightarrow \mathbb{C} ; \exp{(f(z))} = z$ is not continuous using the fundamental group. I´ve found this Does every continuous map induce a ...
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### Proving Existence of Discontinuity

I need to prove that $f:[0, 1] \to \Bbb R$ given by $f(x) = \begin{cases} 1, & \text{if$x=\frac{1}{n}$for any positive integer$n$} \\ 0, & \text{otherwise} \end{cases}$ has an infinite ...
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### What is meant by “discontinuous” here?

I am reading signal processing first by Mcclellan In chap 3,last para of article 3.1.2, I came across a term "discontinuous" as shown underlined in attached photo What is meant by it in the context ...
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### Proving an equivalent characterisation of non existence of left and right hand limit for a bounded function

Here's the lemma that I've been trying to prove: Suppose $f: A \to \mathbb{R}$ is bounded and there is a $\delta >0$ such that $(a, a+\delta) \subset A$. Then $\lim_{x\to a^{+}} f(x)$ fails ...
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### Unique increasing solution to a separable differential equation (piecewise $C_1$)

I want to find the increasing function $y(x): [0,1] \rightarrow [0,1]$ which is defined implicitely as the solution to the following equation: $f_1(x) = f_2(y(x)) \quad \forall x \in [0,1]$ ...
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### absolute asymptotic condition number uniqueness differential equation

Suppose that we have a piecewise, differential equation. Say $\frac{dx}{dt} = \begin{cases} x \sin \frac{1}{x} & x \neq 0 \\ 0 & x = 0\end{cases}$ $x(0) = 0$ I like to ask if the ...
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### What is it called when a function's maximizer is discontinuous in a parameter value?

Suppose I have function: $f(x;a)$ where $a$ is a parameter. The maximizer $x^{*}(a)=\text{argmax}_xf(x;a)$ is discontinuous in $a$. Is there a general term for this? I know this question is vague, ...
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### Is the function $f$ continuous at $(0,0)$?
Does $f$ have a limit as $(x,y)$ tend to $(0,0)$? If yes calculate the limit. $$f(x, y) := \dfrac{1−\cos (xy)}{xy^2}$$ [ Hint: consider the variable $t := xy$ ] So first I showed that the function ...
Discuss whether the given function is smooth, piecewise smooth, continuous, piecewise continuous, or none of these on the interval $\left [ -\pi ,\pi \right ]$ \$f(x)=\left \{ \begin{matrix} 1 & ...