# 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.

221 questions
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### What's the difference between a singular function and a singular continuous function?

I am a physicist so I am trying to make sense of definitions. As far I know, a singular function on $[a,b]$ is defined as: $f$ is continuous on $[a, b]$. the derivative $f′(x)$ exists and is zero ...
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### Prove discontinuity at given point

Given the function $y= \lim_{n \to \infty} \frac{1}{1+x^n}$ for $x \geq 0$, show that the function is discontinuous at $x=1$? I tried the question , it comes out to be continuous using left hand ...
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### A function with discontinuities.

Let $y=f(x)$ be a function which is discontinuous for exactly $3$ values of $x$ but defined $\forall x~{\in}~\mathbb{R}$. Let $y=g(x)$ is another differentiable function such that $y=f(x)g(x)$ is ...
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### Laplace Transform: Piecewise Function Integrability and Existence of Laplace Transform

I am trying to decide whether the function $$f(t) = \begin{cases} 1, & \text{t is even} \\ 0, & \text{t is odd} \end{cases}$$ has a Laplace transform, or is even integrable in the ...
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### Show that $f$ is not Riemann integrable on $[0,1]$

If $x$ is any rational number, $f(x)=0$. If $x$ is any irrational number, $f(x)=1$. I know that $f(x)$ oscillate between $0$ and $1$ on $[0.1]$. But I have not idea why it isn't integrable on $[0.1]$...
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### Example showing $\lim\limits_{x \to x_0} xf(x) \neq x_0\lim\limits_{x \to x_0} f(x)$

I can looking for a simple example to illustrate $\lim\limits_{x \to x_0} xf(x) \neq x_0 \lim\limits_{x \to x_0} f(x)$ For example I have tried $f(x) = x-1, x_0 = 1$ hoping that I would get a zero on ...
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### Any countable set of real numbers is set of discontinuities of some monotone function.

I am studying for a final exam and have come across the following old exam question: Prove that any countable set of real numbers is the set of points of discontinuity of some monotone function. The ...
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### $f:[0,1]\to\Bbb{R}$ has property- $\forall y\in\Bbb{R}$, either $\nexists x\in[0,1]$ s.t. $f(x)=y$ or $\exists$ exactly two such points in $[0,1]$.

My whole question looks like- A real valued function $f:[0,1]\to\Bbb{R}$ has the property that $\forall y\in\Bbb{R}$, either $\nexists x\in[0,1]$ s.t. $f(x)=y$ or $\exists$ exactly two such ...
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### I am not being able to solve this problem of Continuity. [closed]

Prove that $[x] \sin^2(\pi x)$ is continuous at every integer point and $[x] \cos^2 (\pi x)$ is discontinuous at every integer point.
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### Discontinuity - Unsure If Piecewise Equation(s) Have Them

I have a question on whether the functions following have a discontinuity, and if not, what are the points where two functions meet. First, the piece wise equation : \begin{align*} f(x)= \begin{...
I was just thinking a little about functions that map disconnected sets in $\mathbb R^n$ onto connected sets in $\mathbb R^n$. If a function does something like that it seems to me that such a ...