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) ...

learn more… | top users | synonyms (1)

0
votes
1answer
11 views

Monotone functions and distribution functions

I found this quote in a textbook on measure theory I'm studying: Let $f:[a,b] \to \mathbb{R}$ be an increasing function. Since $f$ has only countably many discontinuities, we may assume without ...
1
vote
1answer
24 views

Can a real continuous bounded function on $ \Bbb{R}^{2} $ be expressed as a finite sum of products of real continuous functions on $ \Bbb{R} $?

Can a real-valued continuous bounded function on $ \Bbb{R}^{2} $ always be expressed as a finite sum of products of real-valued continuous functions on $ \Bbb{R} $?
1
vote
1answer
14 views

Charactheristic function approximated by continuous function

I am trying to do the following problem Let $E \subset \mathbb R^d$ be measurable and let $\epsilon>0$. Show that if $A \subset E$ is measurable, then there is $f:E \to \mathbb R$ continuous such ...
0
votes
0answers
12 views

“root” of a right-continuous function

Suppose $f:[0,1] \longrightarrow [-1,1]$ is a right-continuous function such that $f(0) < 0$, $f(1) > 0$, and $f$ only changes sign once in the interval $[0,1]$. Suppose we define the "root" of ...
0
votes
1answer
16 views

Continuity and Directional Derivatives

Does every absolutely continuous function on a compact set possess a left and right hand derivative everywhere on its interior? Although the two need not be equal of course.
2
votes
1answer
38 views

Does the function $f(x) = \frac{x^2-1}{x-1}$ have any point discontinuity?

Since the domain of $f(x)$ is $(-\infty, 1) \cup (1, \infty)$ is there any point discontinuity in $f(x)= \frac{x^2-1}{x-1}$?
0
votes
0answers
26 views
0
votes
0answers
15 views

Limit and Continuity dependency [on hold]

Whether someone knows any interesting example of dependency between limit and continuity of function? Thank you for any proposition
17
votes
5answers
641 views

What is the geometrical difference between continuity and uniform continuity?

Can we explain between ordinary continuity and Uniform Continuity difference via geometrically? What is the best way to describe the difference between these two concepts to someone else? Where the ...
3
votes
1answer
40 views

Show that $f$ is continuous at exactly one point

Let $f:\mathbb{R}\to\mathbb{R}$ be defined by $$f(x)= \begin{cases} 5x+7 & \text{ if } x \text{ is rational } \\ x+11 & \text{ if } x \text{ is irrational } \end{cases}$$ ...
0
votes
1answer
58 views

If $f(U)=0$ then what is possible?

Let , $U=\left(0,\frac{1}{2}\right)\times \left(0,\frac{1}{2}\right)$ and $V=\left(-\frac{1}{2},0\right)\times \left(-\frac{1}{2},0\right)$ and $D$ be the open unit disk centered at origin of $\mathbb ...
2
votes
2answers
39 views

Continuous function without a weak derivative

Let $f:\Omega\to\mathbb{R}$ be a continuous function. Is it necessarily true that $f$ has a derivative in the weak sense? That is, is there some $v:\Omega\to\mathbb{R}$ such that for every test ...
0
votes
0answers
10 views

continuity and convergence [duplicate]

If we have a continuous function that converges on a compact subset of a metric space does it imply that it converges uniformly in general, or is this only in the case if f is monotonic (Dini's ...
2
votes
2answers
27 views

Convergence of a sequence of integrals

I've tried expanding the hinted expression by using the definition from part (i) and choosing an X0 sufficiently large that |f(x)-l| < 1 but this doesn't appear to help very much at all. I've ...
1
vote
2answers
31 views

Continuity and measurability

this question concerns continuity and measureability. Am I right in thinking that if $f>0$ for all x then $\log(f(\lambda x)/f(x))$ is a continuous function for all $x$. Does this then mean that ...
1
vote
2answers
28 views

Continuity of $\mu(t)=\inf\{x \in \mathcal C : \kappa(x)=t\}$.

Let $\Delta = \{ 0, 1\}^{\mathbb N}$ be a Cantor set. Define $\theta : \Delta \to [0,1]$ by the formula $$\theta(x_1,x_2,\dots) = \sum_{n=1}^\infty \frac{2x_n}{3^n}.$$ Denote $\mathcal C = ...
0
votes
0answers
14 views

Let $E$ be the space $L^1(\mathbb{R})\cap L^2 (\mathbb{R})$ equipped with the norm $\|u\|_E = \|u\|_1 + \|u\|_2$.

I am trying to solve this but I got stuck. Help needed. $E$ is a Bananch space. Let $f(x) = f_1(x) + f_2(x)$ with $f_1\in L^\infty(\mathbb{R})$ and $f_2 \in L^2(\mathbb{R})$. Check that the mapping ...
2
votes
0answers
18 views

Continuity of a function in a locally convex topological space

I endow the space of bounded sequences with a locally convex topology $\tau$ such that $\tau$ is strictly finer than the product topology (the topology of pointwise convergence), $\tau_p$, and ...
0
votes
1answer
17 views

Showing that the functional $L[h(x)]=\int_{a}^{b}h(x)f(x)dx$ is continuous

Suppose that we have the functional $L: L^2[a,b] \to \mathbb{R}$ , $L[h(x)]=\int_{a}^{b}h(x)f(x)dx$. $f(x)$ is a well behaving, integrable function in $L^2[a,b]$. I want to show that this is a linear ...
0
votes
1answer
20 views

Can unbounded discontinuous functions be locally bounded?

Consider the function $$f(x) = \frac{x^3}{1+x^3}$$ Obviously this function is discontinuous at $x = -1$ therefore discontinuous on $\mathbb{R}$. Moreover, it is unbounded at the same point. Now, I ...
1
vote
1answer
37 views

Two functions equal in some point

I have two continuous functions $f,g$, $f(0) \lt g(0), f(1) \gt g(1)$. How do I prove without using "advanced" theorem (using only definitions of limit, continious functions and sup/inf definitions), ...
2
votes
2answers
27 views

Rationale behind a proof regarding a continuous function and an open ball

can I have the rationale for the first line of this proof? i.e. How did you know to start answering the question in this manner? I am guessing it is because you want to exploit the definition of ...
0
votes
1answer
18 views

On the definition of uniform continuity over an interval.

I was reading some slides and I stumbled upon this definition of uniform continuity in an interval I am unsure on how to trace this back to the definition of uniform continuity that I know: A ...
1
vote
1answer
22 views

Proving isometry and continuity from a positive definite symmetric real matrix

The following is a question from a previous assignment that I was unable to complete. Any assistance on how to complete this would be appreciated. Let $\epsilon$ be the Euclidean metric on $\Bbb R^n$ ...
-1
votes
0answers
20 views

Continous frunctions problem

The problem says: f,g:[0;1]->[0,1] ,2 continous functions.They have the property that f(g(x))=g(f(x))). To solve: Both having the property of DARBOUX on the interval ,demonstrate that the numbers "c" ...
2
votes
1answer
23 views

Surjectivity of expanding map

Suppose that $(X, d)$ is a compact metric space and that $f: X \rightarrow X$ is a continuous function satisfying $d(x,y) \leq d(f(x), f(y))$ for all $x, y \in X$. Show that $f(X) = X$. Here is a ...
1
vote
1answer
28 views

Proving the continuity of functions from one metric to another

I'm studying mathematics at university, and am having trouble with some of the continuity questions. The following is a question from a previous assignment that I was unable to complete. The original ...
0
votes
1answer
37 views

Continuity of a peculiar function

I came across the question: Evaluate$f(x)=\lim_{m \to \infty}\lim_{n \to \infty}[\cos(n!\pi x)]^{2m}$. I simplified this to: $$f(x)= \begin{cases} 1 & \text{if $x \in Q$} \\ 0 & \text{if ...
1
vote
2answers
27 views

If function has a given limit then to prove that function is bounded.

How to Prove that if a function $f : A \to \Bbb R$ has a limit $l \in \Bbb R$ at $c \in L(A)$, then it is bounded in a neighborhood of $c$, i.e. there exists $M \in \Bbb R$ and $\delta > 0$ such ...
1
vote
1answer
12 views

Let $f:[0,\frac{\pi}{2}]\to R$ be $f(x)=max\{x^2,cosx\}$.Prove $f(x)$ attains minimum at $x_0$ and is a sulution to $x^2=cosx$

I try to write $f(x)=\frac{1}{2}x^2+\frac{1}{2}cosx+\frac{1}{2}|x^2-cosx|$ and use the Extreme Value Theorem to show that $x_0$ exists in $[0,\frac{\pi}{2}]$, but I don't know how to show the seconde ...
0
votes
1answer
15 views

Boundedness Theorem for continuous functions on intervals

Just want to confirm this is a suitable proof: Assume $f$ is not bounded on $I$. So, for any $n \in \mathbb{N}$, $\lvert f(x)\rvert > n$. Since $I$ is bounded, $x_n$ is also bounded. By ...
0
votes
1answer
68 views

$f(x) =\ln(2x^2 + 1)$ is continuous on $\mathbb{R}$

True or False The function $f : \Bbb R \to \Bbb R$ defined by $f(x) = \ln(2x^2 + 1)$ is continuous on $\Bbb R$. I know this condition that The function $f$ is continuous at some point $c$ of its ...
-4
votes
4answers
27 views

Let $f,g$ be continuous from $\mathbb R$ to $\mathbb R$ [duplicate]

Let $f, g$ be continuous from $\mathbb R$ to $\mathbb R$, and suppose that $f(r) = g(r)$ for all rational numbers $r$. Is it true that $f(x) = g(x)$ for all $x \in \mathbb R$?
1
vote
1answer
26 views

I need help finishing this proof using the Intermediate Value Theorem?

Let $f$ and $g$ be continuous functions on $[a,b]$ such that $f(a)\geq g(a)$ and $f(b) \leq g(b)$. Prove $f(x_0)=g(x_0)$ for at least one $x_0$ in $[a,b]$. Here's what I have so far: Let $h$ be a ...
2
votes
0answers
12 views

Test for uniform continuity

Test for uniform continuity the function $ f(x, y) = (x^2 + y^2)^\alpha \sin{\frac{1}{x^2+y^2}} $ in $ \{ x^2+y^2 > 1\} $ If we consider $ \alpha < 1 $, then $ \lim_{\sqrt{x^2+y^2} \to ...
0
votes
2answers
45 views

Expanding a function

Is it possible to expand a function $$ f(x,y) = \dfrac{\sin (xy)}{\sqrt{x^2 + y^2}} $$ so it will be continuous on $\mathbb{R}^2$? Now, the denominator should not be equal to $0$, so for the domain, ...
0
votes
1answer
18 views

Show that if $g((x_n)) \rightarrow l$ and $g((y_n)) \rightarrow m$, then $l=m$

Suppose that $g: (a,b] \rightarrow \mathbb{R}$ is uniformly continuous. Suppose that both $(x_n), (y_n)$ are sequence in $(a,b]$ which converge to $a$. Show that if $g(x_n)) \rightarrow l$ and ...
0
votes
2answers
24 views

Continuous multivariable function without limits in a point

I am curious, if there can be a function $f(x,y)$, which is continuous in a point $[0,0]$, but for which iterated limit $\lim _{x \to 0} \lim _{y \to 0} (f(x,y))$ does not exist. Is it even possible ...
0
votes
1answer
27 views

Show that $f$ is uniformly continuous.

Suppose that $F:(a,b] \rightarrow \mathbb{R}$ is continuous and that the limit as $x \rightarrow a$ of $f(x)$ exists. Show that $f$ is uniformly continuous. I am really struggling with this one. HELP ...
1
vote
1answer
23 views

Counter Example to Tietze Extension Property for Arbitrary Topological Space

Above is my question. My only issue is the final bit! For statements $1.$ and $2.$, the answer is true, since in both cases $Y$ is normal and we know that both metric and compact, Hausdorff spaces ...
1
vote
1answer
15 views

Need Help look at function continuity

Consider the following piece-wise function: $$f(x, y) = xy \frac{x^2-y^2}{x^2+y^2}$$ for $(x,y) \neq (0,0)$ and $f(0,0)= 0$. Discuss the continuity of $f$ at $(0, 0)$. Calculate $\partial f / ...
2
votes
2answers
37 views

A little guidance on finding the limit

How do I find the limit of $f(z) = \frac{x^2y}{x^3+y^3} + ixy$ as $z \to0$ ? What I think is if $z\to0$, that implies $x ,y\to0$. But since the $f(z)$ has both variables $x$ and $y$ mixed together, ...
1
vote
1answer
41 views

Proving the continuity of functions from metrics [closed]

Context: I'm studying mathematics at university, and am having trouble with some of the continuity questions. The following is a question from a previous assignment that I was unable to complete. The ...
0
votes
2answers
24 views

Finding the limit of complex function

I am trying to check the continuity of this complex function at the origin. $f(z)=\begin{cases} \operatorname{Im}( \frac{z}{1+|z|} ) \qquad &\mbox{when } z\neq0,\\ 0 \qquad ...
1
vote
2answers
62 views

Real Analysis: Continuity

$f(x)=\left\{ x^2+x, x \in \Bbb Q\right\}, f(x)=\left\{ x^3 + 1, x \notin \Bbb Q \right\}$ I want to prove that $f$ is discontinuous at $x \ne 1$. What I have so far is: Fix $\delta > 0$. We ...
0
votes
1answer
29 views

finding the continuity of a function

I need to find the value of $a$ for which the function $f(x,y)= \frac{x^2-y^2}{x^2+y^2}$ if $(x,y) \neq (0,0) $ and $f(x,y)=a$ when $(x,y)=(0,0)$ when continuous along the path $y=b\sqrt{x}$ where ...
0
votes
1answer
19 views

Need help in understanding proof of continuity of monotone function

I am reading the following proof of a proposition from Royden+Fitzpatrick, 4th edition, and need help in understanding the last half of the proof. (My comments in italics.) Proposition: Let $A$ be ...
0
votes
1answer
43 views

a continuous function

Let $C([a,b])$ be the collection of all functions $f:\mathbb{R} \to \mathbb{R}$ such that continuous on $[a,b]$. It is known that if $f\in C([a,b])$ then $f$ is continuous on every sub-interval of ...
0
votes
2answers
25 views

If a continuous function is strictly decreasing before a point and strictly increasing afterwards, is the point a global minimum?

I'm in the middle of a proof that a point on a function is a global minimum. Usually I'd just solve an inequality to prove by contradiction that there are no points less than the minimum. But I can't ...
0
votes
1answer
25 views

Continuity of the joint distribution function given continuity of marginals

Suppose $X$ and $Y$ are continuous random variables such that $F_X$ and $F_Y$ are the respective distribution functions. Suppose $F_X$ is continuous at $x_0$ and $F_Y$ is continuous at $y_0$. Then ...