Intuitively, a continuous function is one where small changes of input result in correspondingly small changes of output. Use this tag for questions involving this concept. As there are many mathematical formalizations of continuity, please also use an appropriate subject tag such as (real-analysis) ...

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Multivariable: Continuity of Piecewise function

I have this Multivariable problem.... Where I have to find out if a function is continuous or not. Here is the problem: $f(x, y)=\left\{\begin{matrix} \frac{x^4+3y^4}{x^2+y^2} & (x,y)\neq (0, ...
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1answer
29 views

Use the definition of differentiation on a piecewise function.

I need to find the derivative at $x=0$. $$ f(x)= \begin{cases} x^2\sin(1/x) & \text{if } x\neq 0 \\ 0 & \text{if } x \leqslant 0 \end{cases} $$ Using the definition, I know that it's equal ...
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1answer
39 views

If $f:\mathbb C \to \mathbb C$ is continuous at a point $z_0$, then show that $\overline {f(\bar z)}$ is also continuous at $z_0$.

If $f:\mathbb C \to \mathbb C$ is continuous at a point $z_0$, then show that $\overline {f(\bar z)}$ is also continuous at $z_0$. Is the same true for the differentiability at $z_0$? I'm trying to ...
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2answers
26 views

Continuity and integrability;is it true?

If we have a discontinuous real function of all nonnegative terms and $\int^b_a fdx=0$, then does that necessarily imply $f(x)=0$? I can't come up with an example to help me understand.
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36 views

Discontinuity and differentiation;is this possible?

If $ f $ is a continuous function defined on a real interval that has a discontinuity at a point (but is continuous otherwise), then is it possible to be differentiable at that point?
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21 views

Rotations of the Plane

When is a rotation of a plane not continuous? I know that if I take a point $(x,y)$ then the rotation is $(x\cos(\theta)-y\sin(\theta),x\cos(\theta)+y\sin(\theta))$, but I keep getting that this ...
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31 views

Can we find such a monotone function? [duplicate]

Can we find a monotone function $f:[0,1]\rightarrow\mathbb R$ whose discontinuity set is exactly the set $\mathbb Q\cap [0,1]$? Or can we prove that such a function does not exist?
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28 views

Does having infinite limits at a point of discontinuity imply having a vertical asymptote?

Considering that discontinuities occur at holes, jumps, and vertical asymptotes. Is it possible for a function to have a limit from the left of infinite and the limit from the right - infinite if the ...
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1answer
39 views

Compute the righthand limit; calculus

Let $ f $ be a function defined on a real interval from $0 $ to $1$ and have a discontinuity at $1/2$ (however the righthand and lefthand limits still exist). Let $ F $ be a function defined by $ ...
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0answers
21 views

Continuity of Product Topology [duplicate]

Let $X_1, X_2, Y$ be topological spaces and let $X_1 \times X_2$ be the topological space obtained by furnishing the Cartesian product set with the product topology. Let $f: X_1 \times X_2 \to Y$ be a ...
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3answers
45 views

Prove that a function that maps a discrete metric space to any metric space is continuous [closed]

Let $f:D→M$ where $M$ can be any metric space and $D$ is any set with the discrete metric. Prove that $f$ is continuous. I'm not sure where to begin with this.
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1answer
405 views

Given two potatoes, prove that there is a loop of wire which fits around both

This is a classic problem in geometric continuity and I want to see if there are some solutions other than the one I'm thinking of: Two potatoes are given. Prove that there exists a closed loop of ...
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2answers
99 views

Definition of sequential continuity: converse?

A function $f: \mathbb R \to \mathbb R$ is called sequentially continuuous if $x_n \to x$ implies $f(x_n) \to f(x)$. Every continuous function is sequentially continuous. Let $f$ be a continuous ...
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1answer
18 views

The definition of continuity for linear functionals

I am trying to prove that a linear functional is continuous on the space $H^1(0,l)$, and I have a couple of different definitions. The one that I want to use is that $f$ is continuous if $f$ is ...
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2answers
20 views

Continuity of an increasing function on a dense set

Let $f$ be increasing on $D$ ($D$ is dense in $\mathbb{R}$), and define $\tilde{f}$ on $(-\infty,\infty)$ as follows: $$ \forall x: \tilde{f}(x) = \inf_{x<t\in D} f(t).$$ Show that continuity of ...
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1answer
32 views

Problem in standard proof of continuity when pre-image is open?

I have seen several proofs of the fact that a function $f$ from a metric space $X$ to a metric space $Y$ is continuous if every open set on $Y$ has an open inverse image on $X$. When proving the ...
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0answers
29 views

If f is continuous and f $\in$ L1 then $\lim_{\tau\to \tau_0} \Vert f_\tau-f_{\tau_0}\Vert_{L_1}$ = $0$?

Where $ \Vert f_\tau-f_{\tau_0}\Vert_{L_1}$ = $\int_\mathbb{T} \vert f(t-\tau)-f(t-\tau_0)\vert \ dt$ I find it easy to see when f is uniformly continuous, since we would have $\vert ...
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1answer
42 views

Differentiable implies continuous - in more dimensions?

I know that "differentiable of function f in $x_0$ implies continuous of function f in $x_0$". Can I use the same proof to show that it is valid for a function $f: M \to \mathbb{R}^m$ with $M ...
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0answers
16 views

Differentiability of a upper semicontinuous function

assume I am facing the following function: $$f(x)=ln(x)+\imath_{x\ge y}$$ It is clear that it is upper semicontinuous. But can anyone give me a hint how to see if the derivative is continuous or not ...
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2answers
80 views

Determining whether a function is uniformly continuous

Determine whether $(4x-3)/(x-2)$ is uniformly continuous on the open interval $(1,2)$. I'm not sure how to start this as I have only answered these questions with closed intervals?
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1answer
40 views

Closure of the interior of the epigraph

Suppose $f:E\to(-\infty,\infty]$, where $E$ is a Banach space, is lower semi-continuous, convex, and the interior of $epi(f)\neq\emptyset$. Show that $\overline{int(epi(f))} = epi(f)$ \begin{equation} ...
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1answer
36 views

Prove $f(x) = 1/x$ is continuous at $x = 1/2$

I need to write an $\epsilon, \delta$ proof. I know that $\delta$ must be less than $\frac12$, but I can't figure out the other $\delta$ in terms of $\epsilon$. Thank you in advance!
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40 views

Proving continuity of $x^{2}-2$ with $\epsilon$ and $\delta$

Here is my attempt: $f$ is continuous in $p$ if $\forall \epsilon > 0\; \exists \delta > 0$ such that $|x-p|<\delta \Rightarrow |f(x) - f(p)|<\epsilon$ Then for $f(x)=x^{2}-2$ in $x=p$ ...
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1answer
22 views

Proving extreme value theorem; is showing maximum enough?

Can we prove the extreme value theorem by merely showing that a maximum exists (if $f$ is continuous and defined on a closed, bounded interval in $\mathbb{R}$) because then we'd apply this "half" of ...
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3answers
37 views

Prove or disprove: If $f$ is continuous and differentiable in $[a,b]$ then $a$ is a local minimum or a maximum point in $[a,b]$.

Prove or disprove: If $f$ is continuous and differentiable in the interval $[a,b]$ then $a$ is a local minimum or a maximum point in $[a,b]$ I'm trying to disprove by giving a counterexample, any ...
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1answer
28 views

Examine the continuity of complex function

There is confusion regarding continuity of the following function. When solving in polar form it comes continuous but when solving in $x$ and $y$ then not continuous. Examine the continuity of ...
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2answers
59 views

Existence of Zero Divisors in $C(X,\mathbb{R})$

Consider any topological space $X$ and $\mathbb{R}$ be with usual topology. The set of all continuous functions from $X$ to $\mathbb{R}$, denoted by $C(X,\mathbb{R})$, is a commutative ring with unity ...
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21 views

How to prove that a function's image is closed and bounded, without using Heine-Borel's theorem?

Just wanna apologize for potential mistakes since I've never asked any questions here. I've been trying to solve this one all day long, but did not succeed. I have a continuous function and it's ...
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2answers
48 views

Prove: using the Fundamental theorem of calculus

I have trouble doing proofs using the fundamental theorem of calculus and I think seeing an example would help. Suppose we have a continuous function $ f $ defined on a real interval and a function $ ...
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Continuous second derivative over the support of a Daubechies4 wavelet

I can not entirely follow the proof from section 3.1.1 from the book "A primer on Wavelets" by Walker. After the first part (listed below), I can grasp the rest so if you could help I would greatly ...
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1answer
23 views

How to determine continuity in higher dim

$$f(x,y) = \frac{1-\cos{\sqrt{xy}}}{y}$$ $$f(x,0) = \frac{x}{2}$$ How do I prove this is continuous in the quadrant $x,y \ge 0 $? I can't find counterexamples (weak). I'm just starting working in ...
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1answer
39 views

If $f$ is uniformly continuous on $[1,\infty)$ then $\exists \lim_\limits{n\to \infty}{f(x)}=L$, $-\infty \le L\le \infty$.

Prove or disprove: if $f$ is uniformly continuous on $[1,\infty)$ then there exists a limit, $\lim_\limits{n\to \infty}{f(x)}=L$ where $-\infty \le L\le \infty$. I tried to find a counterexample but ...
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1answer
16 views

Is the function that maps a matrix to the determinant of a submatrix continuous?

Let $M$ be the space of $m \times n$ matrices over $\mathbb{R}$. For each $A$ in $M$ let $A'$ be a fixed submatrix of $A$. Is the function $M \to \mathbb{R}$ defined by $A \mapsto \det(A')$ ...
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1answer
22 views

A confusing discontinuity proof using eps-delta

So $f(\vec{x}) =0$ if $x=0$ and equals $xyzt/(x^4+y^4+z^4+t^4)$ if $|\vec{x}|$ does not equal zero. How do I prove it is not continuous at the origin with epsilons and deltas? The whole epsilon delta ...
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Shouldn't this function be discontinuous everywhere?

I was thinking about single point continuity and came across this function. $$ f(x) = \left\{ \begin{array}{ll} x & \quad x\in \mathbb{Q}\\ 2-x & \quad x\notin ...
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1answer
24 views

$\lim_{z\to 0}f(z)$, where $f(z)=\frac{xy^3}{x^3+y^3}+\frac{x^8}{y^2+1}i$.

$\lim_{z\to 0}f(z)$, where $f(z)=\frac{xy^3}{x^3+y^3}+\frac{x^8}{y^2+1}i$. I'm trying to find the limit of this function. I've tried several directions, $x=y, x=y^2, etc$ but they all ended up with ...
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1answer
26 views

Proof of a corollary of the Banach Fixed Point Theorem

If $(X,d)$ is a complete metric space, and $f: X \rightarrow X$ is a continuous function, show that if $f^{N}$ is a contraction (for some $N > 0$),then $\exists! x \in X$ such that $f(x) = x$. I ...
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3answers
33 views

Are the Unit Ball and Any other Ball Topologically Equivalent

How would I correctly show that the unit ball $B(0,1)\subset \mathbb{R}^n$ and the ball $B(a,r) \subset \mathbb{R}^n$ are Topologically Equivalent? I know I need to find a one-to-one function $f: ...
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2answers
27 views

Show the following function is continuous

Let $ f $ be defined by $f(x)=x$ if $x>0$ and $f(x)=0$ if $x\le 0$. I think it's obvious that it's continuous but maybe it still needs to be shown. What is the easiest way to do this?
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60 views

Computing $\lim_{(x,y)\to (0,0)}\frac{\sin(x+y)}{x+y}$

I'm trying to compute the following limits and the textbook that I'm looking at suggested the following method. $$\lim_{(x,y)\to (0,0)}\frac{\sin(x+y)}{x+y}$$ $$\lim_{(x,y)\to ...
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11 views

Prove the transversal crossing property

I have the following question: Let $x: R$ $\rightarrow$ $R$ be a continuously differentiable function. Assume that $x(0)$ = $0$. Prove that there exists $\delta$ $> 0$ such that $x(t) \neq 0$ for ...
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52 views

Is pointwise maximum absolutely continuous?

Consider two absolutely continuous functions in an interval $I$, $f(x)$ and $g(x)$. Is the pointwise maximum, $\max(f(x),g(x))$, also absolutely continuous?
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1answer
35 views

Show that $f(z)=1/z^2$is not uniformly continuous for $0\lt Rez\lt 1/2$ but is uniformly continuous for $1/2\lt Rez\lt 1$.

Show that $f(z)=1/z^2$is not uniformly continuous for $0\lt Rez\lt 1/2$ but is uniformly continuous for $1/2\lt Rez\lt 1$. To prove the first assertion, I came up with sequences $x_n=1/n, ...
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2answers
47 views

Homeomorphism from $(-1,1)$ to $\mathbb R$

I know that $f: (-1,1) \to \mathbb R$ defined by $f(x)=\tan \Big(\dfrac{\pi}2x \Big)$ is a homeomorphism . I am looking for some other homeomorphism between $(-1,1)$ and $\mathbb R$ which is not in ...
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16 views

Continuous function generating continuous angle function

Let $f,g:I\to\mathbb{R}$, where $I\subseteq\mathbb{R}$ is an open interval, be two continuous functions. Show that there is a continuous function $\theta: I\times I\to\mathbb{R}$ such that: ...
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1answer
37 views

Measure Theory: Continuity

How can I find a subset of a set with "half the size" of the original? I am trying to solve this problem and I came across this post. The solution using the intermediate value theorem is ...
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2answers
36 views

Seeking to prove Continuity of $f(x) =\frac{x}{1+||x||}$

How would I prove that $f:\mathbb{R}^n\rightarrow B(\theta,1)$, where $f(x)=\frac{x}{1+||x||}$, is continuous? For metric spaces, I understand that if $f(x)$ is continuous at a point $p$ ...
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32 views

Proving that a rational function with two variables is continuous

for some reason i'm struggling with this very basic propsition: Let $f:\mathbb R \times \mathbb R \to \mathbb R$ be a rational function. for that matter we can every assume $f(x,y)=\frac{1}{p(x,y)}$ ...
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2answers
102 views

Calculate limit for continuous function

I am having trouble with this question: if f(x) is a continous function and $$ 1 - x^2 \leq f(x) \leq e^x $$ for all values of x, calculate $$ \lim_{x\to0} f(x) $$. Do I have to calculate the limit ...
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2answers
31 views

Let $f$ have a jump discontinuity at $x_0$. Show that $f(x_1), f(x_2), \ldots$ has at most two limit-points.

This is a question I understand intuitively but am having trouble proving rigorously: Let $f$ have a jump discontinuity at $x_0$. Show that if $x_1, x_2, . . .$ is any sequence of points in the ...