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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Help with proof of theorem 1-10 spivak calculus on manifolds

Theorem 1-10 The bounded function (mapping to $\mathbb{R}$), $f$ is continuous at $a$ if and only if the oscillation of a point of $f$ at $a$ is $0$ The definition spivak provides of oscillation ...
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1answer
46 views

Is every differentiable function on $(0,1)$ uniformly continuous $?$

$$f:(0,1)\rightarrow [0,1]$$ is a differentiable function . Is it uniformly continuous then $?$ Now $f$ being differentiable on $(0,1)$ is continuous , that is easy. Now I could ...
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1answer
22 views

Showing value exists with intermediate-value theorem

I have this problem: "Show that the function $f(x) = (x-a)^{2}(x-b) + x$ has a value $f(c) = \frac{a+b}{2}$ for a number c" I am new to this kind of problems and I am having a bit of trouble ...
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3answers
35 views

Looking for example of topological spaces where sequential continuity does not imply continuity

Please give an example of a function $f : X \to Y $ where $X,Y$ are topological space , such that there exist $x \in X$ such that for every sequence $\{x_n\}$ in $X$ converging to $x$ , $\{f(x_n)\}$ ...
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5answers
76 views

Which of the following condition ensure that the function $f:R^n\to R$ is continuous?

I encountered an interesting problem in my Economics class about continuity. Which of the following conditions on the function $f:\mathbb R^n\to \mathbb R$ ensures that the function $f$ is ...
2
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3answers
110 views

$f:\mathbb R^{2} \rightarrow \mathbb R$ s.t ${f(x,y)}={{xy}\over {x^{2}+y}}$ is not continuous at the origin

$f:\mathbb R^{2} \rightarrow \mathbb R$ is defined as $${f(x,y)}={{xy}\over {x^{2}+y}}$$; when $x^{2}+y\neq 0$ and $$f(x,y)=0$$ otherwise. To show this is not continuous at the origin . ...
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1answer
17 views

Continuity of the following complex function

I am working through the text Complex Analysis by George Cain and I have a question stemming from the problem #9 in section 2.2. Consider the function $f(z)$ given by $\frac{\overline{z}^2}{z}$ when ...
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0answers
25 views

Is the function ${e^{-{1}\over{x}}}\over {x}$ on $(0,1)$ uniformly continuous or bounded?

$$f(x)= {{e^{-{1}\over{x}}}\over {x}}$$ for $x\in (0,1)$ . Is this function $a$) uniformly continuous $b$) bounded but not continuous $c$) unbounded This would be uniformly ...
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1answer
41 views

Is $\sqrt{x}$ Lipschitz continuous on $(0,\infty)$?

Is $\sqrt{x}$ Lipschitz continuous on $(0,\infty)$? I wonder that because I know it is on $[a,\infty)$ for all $a>0$.
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2answers
54 views

$f: X \to Y$ is continuous if and only if $f: X \to f(X)$ is continuous

Let the image of $f$, which is $f(X)$, have the subspace topology inherited from $Y$. Prove that $f: X \to Y$ is continuous if and only if $f: X \to f(X)$ is continuous. If $f: X \to Y$ is ...
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1answer
32 views

Understanding a proof of the Pasting Lemma

Pasting Lemma. Let $A$ and $B$ be both open or closed subsets of a topological space $X$ such that $A \cup B = X$. Let $f: A \to Y$ and $g: B \to Y$ be continuous such that $f = g$ for all $x \in A ...
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2answers
40 views

Limit of an integral of a continuous real-valued function

If $f:[0,{\infty})\to\mathbb R$ continuous and $\lim_{x\to\infty} f(x)=a$. Show that: $$ \lim_{x\to\infty} \frac1x\int_{0}^{x} f(t)\ \mathsf dt = a. $$ If: $$ \lim_{x\to\infty} \frac1x ...
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0answers
36 views

Continuity at $(0,0)$ of $f(x,y)=2xy^2/(x^2+y^4)$ along the paths $φ(t)=(t,t)$ and $ψ(t)=(t^2,t)$

Let $f: \mathbb R^2→\mathbb R$, $φ: \mathbb R→\mathbb R^2$, $ψ: \mathbb R→ \mathbb R^2$ be given by $φ(t)=(t,t)$, $ψ(t)=(t^2,t)$, $t ∈ \mathbb R$ and $$f(x,y) = \begin{matrix} \frac{2xy^2}{x^2+y^4} ...
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1answer
27 views

Is f continuous t (0,0) for any choice of K? Explain fully.

Let f: $R_2$ $→\mathbb R,φ: \mathbb R→R_2, \, ψ: \mathbb R→ \mathbb R2$ be given by φ(t)=(t,t),ψ(t)=($t^2,t$),t \, t ∈ $\mathbb R$ and $$f(x,y) = \begin{matrix} \frac{2xy^2}{x^2+y^4} \quad \text{if ...
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2answers
31 views

When can interchange taking a limit with applying a function?

I've seen some limit problems where you can do this: $$ \lim_{x \to \infty} \exp\left({g(x)}\right) = \exp \left( \lim_{x \to \infty }g(x)\right) . $$ So, I've tried to generalize the result as: $$ ...
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0answers
31 views

Proving “if” direction of continuous iff sequence x_n converging to x implies f(x_n) converges to f(x)

Here is the theorem in mathjax: A real value function $f$ is continuous at $x \in R$ iff whenever a sequence of real numbers $x_{n}$ converges to $x$ then the sequence $f(x_{n})$ $\rightarrow f(x)$. ...
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16 views

$f:[a,b] \to [0, \infty)$ continuous , then $\lim_{n \to \infty} \Bigg(\int_a^b \big(f(x)\big)^ndx \Bigg)^{1/n}=\sup \{f(x):x \in [a,b]\}$ ? [duplicate]

Let $f:[a,b] \to [0, \infty)$ be continuous , then is it true that $\lim_{n \to \infty} \Bigg(\int_a^b \big(f(x)\big)^ndx \Bigg)^{1/n}=\sup \{f(x):x \in [a,b]\}$ ?
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12 views

Single-statement Continuous Periodic function without trigonometry and complex numbers

Superseding the question Periodic function without trigonometry and complex numbers , I am now asking: Can I get a single-statement continuous periodic function without using trigonometric functions ...
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23 views

continuity of the piecewise functions [closed]

$1$. $g(x)=0$,if $x$ is irrational and $g(x)=x$ if $x$ is rational Find all points of at which $f$ is continuous. $2$. Let $A$ and $B$ be compact sets. Define $A+B =$ ...
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1answer
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+300

Intuition for visualising dense monotonic discontinuous function

My question is about the function defined in Rudin 4.31, mentioned by this question: Remark 4.31 in Baby Rudin: How to verify these points? The function is defined as $$f(x) \colon= \sum_{x_n < ...
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4answers
192 views

Is there a continuous function from $[0,1]$ to $\mathbb R$ that satisfies

Is there a continuous function $f:[0,1] \to \mathbb R$ such that $f(x) = 0$ uncountably often and, for every $x$ such that $f(x) = 0$, in any neighbourhood of $x$ there are $a$ and $b$ such that $f(a) ...
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2answers
43 views

$f \in C[a,b]$ be such that $\int_c^d f(x)dx=0 , \forall c,d \in [a,b] , c<d$ ; then $f$ is identically zero on $[a,b]$?

Let $f:[a,b] \to \mathbb R$ be a continuous function such that $\int_c^d f(x)dx=0 , \forall c,d \in [a,b] , c<d$ ; then is it true that $f(x)=0 , \forall x \in [a,b]$ ?
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2answers
31 views

Proof weird function is discontinuous/has no partial derivatives.

I'm asked to analyze the continuity and existence of partial derivatives at the origin, and even though it seems pretty obvious that this function is discontinuous at that point, I can't seem to prove ...
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1answer
22 views

Help with two functions - continuity, Laplace transform and Fourier series [closed]

I've been practicing for my exam lately, and there are two function that I've had a real trouble analyzing. 1.$f(x) = \sum_{n=1}^{\infty} \frac{\sin(nx)}{10^n \sin(x)}$, for $x \neq k\pi$ $f(x) = ...
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2answers
32 views

CPT exam Quantitative aptitude exercise 8c [closed]

$$\lim\limits_{n\to \infty}\left[\frac{1}{6}+\frac{1}{6^2}+\frac{1}{6^3}+\cdots+\frac{1}{6^n}\right]$$ is: (a) $\frac15$ (b)$\frac16$ (c)$-\frac{1}{5}$ (d) none of these According to book ...
2
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1answer
40 views

If a linear operator between two normed linear spaces is continuous at one point, then it is continuous at all points.

Let $f : \langle V_1, \|\cdot\|_1\rangle \to \langle V_2, \|\cdot\|_2\rangle$ be linear. Then if $f$ is continuous at some $v \in V_1$, then it is continuous on all of $V_1$. Without appealing to ...
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0answers
22 views

Check whether the sum of the series $\sum^{\infty}_{n=1}\frac{\sin(nx)}{nx}\cos\frac{x}{n}$ is continous on $(0,\pi)$

Check whether the sum of the series $\sum^{\infty}_{n=1}\frac{\sin(nx)}{nx}\cos\frac{x}{n}$ is continuous on $(0,\pi)$ I think about showing the uniform convergence of $$ f_k: \mathbb (0, \pi) ...
3
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3answers
49 views

Why does $\lim_{x\to 0} \frac {\sin (xy)}{x} \to y $?

Let $f(x,y) = \frac{\sin (xy)}{x}$ for $x\neq 0$. How should you define $f(0,y)$ for $y\in \mathbb{R}$ so as to make $f$ a continuous function on all of $\mathbb{R}^2$? So in order for a function to ...
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0answers
23 views

Question concerning continuity of composite functions

Consider two functions, $a(r)$ and $b(r)$. If a is continuous at $c$,and $b$ is continuous at $a(c)$ , then $b(a(c))$ is continuous at $c$ .(This is a theorem stated in the text Thomas' Calculus) Now ...
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0answers
57 views

Give an example of a function $f :X \to Y$ which is sequential continuous but not continuous where $X$ and $Y$ are some topological spaces.

Give an example of a function $f :X \to Y$ which is sequential continuous but not continuous where $X$ and $Y$ are some topological spaces. I have seen some example which uses $X$ to be non ...
1
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4answers
58 views

Show that the function is continuous

To show that the function $f: \mathbb{R}^2 \rightarrow\mathbb{R}$ with $f=\left\{\begin{matrix} \frac{x^3-y^3}{x^2+y^2} & , (x,y) \neq (0,0)\\ 0 & , (x,y)=(0,0) \end{matrix}\right.$ is ...
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3answers
53 views

Intuitive meaning of the probability density function at a point

I understand how to integrate probability density functions to find probability within a certain range. However, what I don't understand is what it would mean to set the variable (say x or y) to a ...
2
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1answer
40 views

Fundamental solution Laplace-Poisson equation

Let $\Phi:\mathbb{R}^n\setminus\{0\}\rightarrow\mathbb{R}$ be the fundamental solution of the Laplace equation (see e.g. in the book of Evans). For a function $f\in\mathcal{C}_c^2(\mathbb{R}^n)$ we ...
0
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1answer
45 views

Is this function bounded or not?

$f(x) = \left(1-\frac ax\right)^2$ where both $x>0$, $a>0$ Is this function bounded? i.e. is there an M such that $f(x) ≤ M < \infty$ ? How can I figure this out? Thanks very much in ...
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1answer
20 views

Piecewise $C_1$ and piecewise continuous

I would appreciate if the following questions could be clarified with your help. If a function is piecewise $C_1$, does this imply that it's also piecewise continuous? If a function is piecewise ...
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2answers
61 views

Explain why continuity along straight lines is not enough to conclude continuity

Consider the function with domain $A = \{ (x,y) \in \, \mathbb{R}^2: (x,y) \neq (0,0)\}$ given by $$\frac{2x^2y}{x^4+y^2}$$ Letting $(x,y)$ approach $(0,0)$ along the straight line $y=ax$ , where ...
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0answers
36 views

Attaining a maximum without applying the Weierstrauss Theorem

The example that mookid gave in this question is a good one. There is no maximum since it is not continuous. How could you explain why the function given by mookid will attain a maximum on any compact ...
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1answer
32 views

A function that satisfies the Intermediate Value Theorem and takes each value only finitely many times is continuous.

I'm having a confusion over the veracity of the statement that a function that satisfies the Intermediate Value Theorem and takes each value only finitely many times is continuous. I've seen from a ...
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2answers
55 views

Show that the mapping $f → f~'$ from $C^1([0 , 1])$ to $C([0 , 1])$ is not continuous.

Let $C^1([0 , 1])$ be the subspace of $C([0 , 1])$ consisting of the functions that have a continuous derivative throughout $[0 , 1]$. Show that the mapping $\Psi:f → f~'$ from $C^1([0 , 1])$ to $C([0 ...
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1answer
42 views

$f\in BV[a,b]$ has the intermediate value property , then is it true that $f$ is continuous on $[a,b]$ ?

If $f\in BV[a,b]$ has the intermediate value property , then is it true that $f$ is continuous on $[a,b]$ ? Please help . Thanks in advacne
2
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1answer
39 views

$f:[a,b]\to \mathbb R$ is continuous , has a finite number of local maxima and minima ; then how to prove that $f$ is bounded variation on $[a,b]$ ?

If $f:[a,b]\to \mathbb R$ is a continuous function having finite number of local maxima and minima ; then how to prove that $f$ is bounded variation on $[a,b]$ ?
4
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0answers
68 views

A discontinuous function $f: X \rightarrow Y$ satisfying: for each closed ball $B$ of $Y, f^{-1}(B)$ is closed in $X$

Find a function $f: X \rightarrow Y$ between metric spaces $X$ and $Y$ that is not continuous but has the property that for each closed ball $B$ of $Y, f^{-1}(B)$ is closed in $X$ Solution Attempt: ...
2
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1answer
67 views

Value of $x \log x$ at $x=0$ [duplicate]

What is the value of $f(x) = x \ln (x)$ at $x= 0$? Is it $0$ or indeterminate?
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0answers
29 views

Continuity of Function involving logarithm function

I want to prove a function $f(x) = g(x) * log x $ is continuous on interval $[0, 1]$, where value of $g (x)$ is $0$ at lower limit point $0$. Anybody can help me out here. Thanks in advance.
1
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1answer
18 views

How to check F:AxI->B is continuous

A and B are topological spaces.Let f and f' are continuous maps from A to B and homotopic.Then we need F:AxI->B,continuous,where F(s,0)=f(s) and F(s,1)=f'(s). Now my question is if we want to ...
1
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1answer
21 views

Average integral for continuous functions with compact support

Let $f$ be a continuous function with compact support in $\mathbb{R}^n$. Show that \begin{equation} \lim_{r\to 0} \frac{1}{|B_r(x)|} \int_{B_r(x)} f(y)\,dy = f(x), \end{equation} where $B_r(x)$ is the ...
2
votes
3answers
48 views

When the set of $r$-far interior points from a set is open

Let $E$ be a subset of a metric space $X$ and for $r > 0$ let $$ E_r = \lbrace x \in E : d(x,E^c) > r \rbrace .$$ Is the set $E_r$ always open? Equivalently, is the function $ x \mapsto ...
1
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1answer
13 views

Proof that a derivative's points of discontinuity are all essential

I'm reading Wikipedia's article on Darboux's theorem, and it says the following: "Every discontinuity of a Darboux function is essential, that is, at any point of discontinuity, at least one of the ...
-4
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1answer
35 views

Prove or disprove about isomorphic functions [closed]

Prove that : if f is an isomorphic then it is continuous or not?
2
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1answer
58 views

Prove that the set of square matrices $A(x)=\begin{pmatrix} 2x+y & x \\ 3x & 2x+3y \\ \end{pmatrix}$ for $x,y\in [0,1]$ is a compact set.

Prove that the set of square matrices $A(x)=\begin{pmatrix} 2x+y & x \\ 3x & 2x+3y \\ \end{pmatrix}$ for $x,y\in [0,1]$ is a compact set.(Take into consideration metric $d_2...$) I was ...