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) ...
1
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2answers
38 views
Are absolute extrema only in continuous functions?
The Extreme Value Theorem says that if $f(x)$ is continuous on the interval $[a,b]$ then there are two numbers, $a≤c$ and $d≤b$, so that $f(c)$ is an absolute maximum for the function and $f(d)$ is an ...
2
votes
1answer
47 views
Continuity of an $\mathbb {R}^2$ function
Let $f$ be an $\mathbb{R}^2$ endomorphism and $N:\mathbb{R}^2\to\mathbb{R}^+$
defined by $$\forall u \in \mathbb {R }^2, N(u) = ||f(u)|| $$
I need to show $N$ is continuous.
The problem is that $N$ ...
2
votes
2answers
41 views
Continuous Linear Mapping $C[0,1]\rightarrow C[0,1]$
Show that $L(f)(x)= \int_0^x f(t) dt $ is a continuous linear mapping from $C[0,1]$ into itself.
Do I only have to show that the operator is bounded? How to do I explicitly choose my $M$ such that ...
5
votes
2answers
82 views
Properties of Continuous Functions
Prove that there is no continuous function $f:\mathbb{R} \rightarrow \mathbb{R}$ such that for $c \in \mathbb{R}$ the equation $f(x)=c$ has exactly two solutions.
This is what I have so far.
Proof by ...
2
votes
3answers
52 views
Show that f is uniformly continuous
I am having difficulty determining what value I should assign to $\delta$ for the following problem. How do I determine what it should be?
Define $f:[3.4,5] \rightarrow \mathbb{R}$ by ...
0
votes
1answer
44 views
Continuous function on a closed set
Let $f: F \to \mathbb R$ be defined in a closed set $F \subset \mathbb R$. Show that $f$ is continuous if and only if for all $c \in \mathbb R$, the sets $E[f \le c]=\{x \in F; f(x) \le c\}$ and $E[f ...
1
vote
2answers
42 views
Definition of Lebesgue-Stieltjes measure on $\mathbb R$
Let $F:\mathbb R\to\mathbb R$ be a non-decreasing, left-continuous function. Let $a,b\in\mathbb R$, then define the Lebesgue-Stieltjes measure
$$ m[a,b]=F(b+)-F(a), \quad m(a,b)=F(b)-F(a+) $$
...
1
vote
0answers
10 views
continuity set of a composite function
Consider $g(.)$ as a continuous and bounded function. Consider $f(x)$ such that $Pr\{X \in C(X)\} = 1$ where $C(X)$ is the continuity set of f.
How can I show that the continuity set of $f(.)$ is a ...
2
votes
0answers
56 views
Weakly closed subsets of $C(K)$
Given a compact Hausdorff space $K$, let us endow $C(K)$ with the Banach-space weak topology. Is there any handy description of weakly closed subsets of $C(K)$? Are subsets of $C(K)$ which are ...
1
vote
0answers
53 views
Let$ f : [0, 1]^2 \to R$ such that $f(x, y)$ is continuous in $x$ for each fixed $y$ and conversely also. Is $f $ continuous?
Let$ f : [0, 1]^2 \to R$ such that $f(x, y)$ is continuous in $x$ for each fixed $y$ and continuous in $y$ for each fixed x. Does it follow that $f$ is continuous?
-1
votes
0answers
28 views
Continuity of quotient map
$f:X\to X/\mathord{\sim}$ is continuous for any space $X$ and equivalence relation $\sim$, where $f$ is defined to be $f(x)=[x]$, and $[x]$ is the equivalence class of $x$.
How can I prove this?
1
vote
3answers
46 views
If $x \leq g(x) \leq x^2-x+1$ where $x \in [0,2]$, can we say that $g(x)$ is continuous at $x=1$?
If $x \leq g(x) \leq x^2-x+1$ where $x \in [0,2]$, can we say that $g(x)$ continuous at $x=1$ ?
Is $g(x)$ continuous in $[0,2]$?
2
votes
1answer
22 views
Sequence of continuous functions, integral, series convergence
Let $f_k$ be a sequence of continuous functions on $[0,1]$ such that $\int _0 ^1 f_k(x)x^ndx = \int _0^1 x^{n+k} dx$ for all $n \in \mathbb{N}$.
Is $\sum _{k=1} ^{\infty}f_k(x)$ convergent?
Could ...
2
votes
2answers
57 views
Proof f(x) is continuous given $x$ rational and irrational.
How can I resolve the task below:
Given $f(x)=
\begin{cases}
x, &x\in \mathbb{Q}\text{ }\\
1-x, &x\notin \mathbb{Q}\text{ (irrational)}
\end{cases}$, $0 \leq x \leq 1$.
Show $f(x)$ is ...
0
votes
1answer
29 views
Continuous variable defined over Rational numbers only?
Let $x(t)$ be a solution of some first order ODE, which is continuous over $t$. In this case, is the continuous $x(t)$ defined only over Rational numbers? what is the reason behind this? Please ...
0
votes
1answer
29 views
continuity and differentiability and L'Hopital's Rule
Let
$$f_n(x) = \begin{cases}
0 & x < -\tfrac{1}{n} \\
\tfrac{n}{2} & -\tfrac{1}{n} \leq x \leq \tfrac{1}{n} \\
0 & x>\tfrac{1}{n} \\
\end{cases},$$
$n=1,2,3,\ldots$.
Let $g(x)$ be a ...
3
votes
1answer
52 views
Proving that a function is differentiable and equal to a constant value for all x
Let $f(x)$ denote a strictly positive continuous function defined on all real numbers with the property that $f(2012)=2012$ and $f(x)=f(x+f(x))$ for all $x$. Prove that $f(x)=2012$ for all $x$.
I am ...
1
vote
2answers
18 views
Relation between continuity and connection between topologies
If $X$ is a set and $\tau_1,\tau_2$ two topologies on $X$. What does it mean to put the continuity of the identic map on $X$ (i.e $id_X(x)=x\forall x\in X$) in a relation to the comparability fo two ...
9
votes
1answer
84 views
Homeomorphism between open unit ball and $\mathbb R^n$
Let $B=\{x\in\mathbb R^n : ||x||<1\}$ the open unit ball with the subapce topology of $\mathbb R^n$. I want to show that $B^n\cong\mathbb R^n$ with the map $F(x)=\tan(\frac{\pi ...
2
votes
0answers
54 views
Proof on showing F(x,y) is continuous by $\epsilon - \delta$ definition
The task is as follows:
Given: $$F(x,y) = \frac{xy(x^2 - y^2)}{x^2 + y^2}$$
Goal: Prove that $F(x,y)$ is continuous everywhere on the plane
Here is my attempt so far:
(1) By the ...
2
votes
1answer
73 views
$\int_0^1\frac{(f(x)-1)^2 -4x^2}{x^{3.5}}\,dx$ exists. Calculate $f(0)$ and $f'(0)$
I've tried somehow using Taylor to try and figure this one out.
Unfortunately, I couldn't seem to get a solid answer.
Thank you very much for your help!
Let f be a continuos function,
...
1
vote
1answer
31 views
How to show $\frac{P(x)}{Q(x)}=\sum_{k=1}^n \frac{Q(x_k)}{P'(x_k)(x-x_k)}$ when the following condition holds?
Let $P$ be a polynomial of degree $n$ with n different real roots
$x_1,x_2,....x_n$ and let $Q$ be a polynomial of degree at most $n -1$. How to Show
that
$ \ \ \frac{P(x)}{Q(x)}=\sum_{k=1}^n ...
3
votes
1answer
39 views
If $f^2(b) - f^2(a) = b^2 - a^2$, then the equation $f'(x)f(x) = x$ has at least one root in (a, b).
Suppose $f$ is continuous on [a, b] and differentiable on the open
interval (a, b). How to that if $f^2(b) - f^2(a) = b^2 - a^2$,
then the equation
$f'(x)f(x) = x$
has at least one root in (a, ...
0
votes
4answers
24 views
How to show that $f(x) = o{(x^2)}$ as $x \to \infty$. when $f$ is differentiable on $(0, \infty)$ and $f'(x) = o(x)$ as $x\ \to \infty$
Suppose $f$ is differentiable on $(0, \infty)$ and $f'(x) = o(x)$ as
$x \to \infty$.
How to show that $f(x) = o{(x^2)}$ as $x \to \infty$?
6
votes
1answer
75 views
Existence of a power series converging non-uniformly to a continuous function
I am wondering whether there exist a function $f(z) = \sum_{n\geq0} a_n z^n$ such that:
$f$ converges and is continuous on the closed unit disk $D$ and
the series $\sum_n a_n z^n$ does not converge ...
1
vote
1answer
31 views
If $f(x)=\sqrt{1-\sqrt{1-x^2}}$, then prove that $f(x)$ is continuous on $[-1,1]$ and differentiable on $(-1,0) \cup (0,1)$.
If $f(x)=\sqrt{1-\sqrt{1-x^2}}$, then prove that $f(x)$ is continuous on $[-1,1]$ and differentiable on $(-1,0) \cup (0,1)$.
Please prove using $$\lim_{x\to c^+}f(x)=\lim_{x\to c^-}f(x)=f(c)$$
...
0
votes
1answer
17 views
If $f(x)=a |\sin x| + b e^{|x|}+c|x|^3$ and if $f(x)$ is differentiable at $x=0$, find the values of $a,b$ and $c$.
If $$f(x)=a |\sin x| + b e^{|x|}+c|x|^3$$ and if $f(x)$ is differentiable at $x=0$, find the values of $a,b$ and $c$.
Please note that if a function is differentiable at a point , it is also ...
2
votes
2answers
31 views
Examine the continuity of $f(x)=x^2+\frac{x^2}{(1+x^2)}+\frac{x^2}{(1+x^2)^2}+…+ \frac{x^2}{(1+x^2)^n}+…$ at $x=0$
Examine the continuity of $$f(x)=x^2+\frac{x^2}{(1+x^2)}+\frac{x^2}{(1+x^2)^2}+...+ \frac{x^2}{(1+x^2)^n}+....$$ at $x=0$
I tried to solve the problem using $$\lim_{x \to0^+}f(x)=\lim_{x ...
0
votes
1answer
22 views
Examine the continuity and differentiability of $f(x)=\frac{\sin (\pi[x-\pi ])}{4+[x]^2}$
Prove that the function $$f(x)=\frac{\sin (\pi[x-\pi ])}{4+[x]^2}$$ where $[.]$ denotes the greatest integer function, is continuous as well as differentiable for all $x \in \mathbb R$
I tried to ...
0
votes
3answers
46 views
Check the continuity and differentiability of $f(x)= \sin^{-1}(\cos x)$ at $x=0$
Check the continuity and differentiability of $f(x)= \sin^{-1}(\cos x)$ at $x=0$
This is how I tried to solve the problem:
$$f(x)= \sin^{-1}(\cos x)=???$$
$$\lim_{x \to 0^+}f(x)=\lim_{x \to ...
0
votes
1answer
32 views
Examine the continuity and differentiability of $f(x)=| \cos x|$
Examine the continuity and differentiability of $f(x)=| \cos x|$
I got the answer to be everywhere continuous and not differentiable
at $ x=(2n+1) \frac{\pi}{2}, n \in \mathbb Z $ (since modulus ...
1
vote
0answers
42 views
How to prove $C^1$ class is a proper subset of Lipschitz class?
Let $Lip(A)$ be the set of vector-valued functions $f$ on the closed set $A\in\mathbb R^n$ such that
$$f(0)=0,$$
$$||f|| \text{ is finite, where by definition: } ||f||=\sup ...
4
votes
2answers
46 views
“Deformation” of the kernel of a linear map
It is known that the roots of a monic polynomial of fixed degree vary continuously (smoothly?) with its coefficients, at least over $\mathbb{C}$. My question is whether there is such a result for ...
0
votes
1answer
40 views
real analysis: continous
Let $g$ be an increasing function on $[a,b]$ to $\mathbb{R}$ and suppose that for each $t ∈[c,d]$, the integral $$F(t) = \int_{a}^{b}f(x,t)\,dg(x) $$ exists
Show that if $f_t$ is continuous on ...
6
votes
2answers
94 views
Relationship between connectedness and continuity
Let $f:\mathbb R^n\to \mathbb R$.
$f$ is continuous,
The graph of $f$ if connected in $\mathbb R^{n+1}$
We define "connected" to be cannot be separated by 2 disjoint non-empty open set.
My ...
1
vote
1answer
20 views
Equicontinuity and uniform convergence 2
Let $\{f_n\}_n$ be a sequence of real valued functions on a compact metric space $K$. Suppose that for all $x$ we have $f_n(x) \to f(x)$ as $n \to \infty$ and that the family $\{f_n\}_n$ is ...
2
votes
0answers
25 views
Discontinuities of monotone operators on arbitrary spaces
Let $X$ be a vector space equipped with an inner product $\langle .,.\rangle$.
A function $f:X\rightarrow X$ is said to be monotone if, for all $x,y$, $\langle f(x)-f(y),x-y\rangle\geq 0$.
On ...
2
votes
3answers
61 views
Necessary and sufficient conditions for differentiability.
Apologizes if I'm missing something in my question or if my question seems trivial; this is my first question on this site. As motivation for my question, consider the following standard first year ...
2
votes
4answers
124 views
example of a function $f :\mathbb{R} \to \mathbb{R}$ whose set of points of discontinuity is $\mathbb{Q}$
I need an example of a function $f :\mathbb{R} \to \mathbb{R}$ whose set of points
of discontinuity is $\mathbb{Q}$.
4
votes
3answers
38 views
Misunderstanding with the local definition of continuity
I know that for any two top spaces $(X,\tau_X),(Y,\tau_Y)$ a function $f:X\to Y$ is said to be continuous on $X$ if $f^{-1}(V)\in\tau_X~\forall~V\in\tau_Y.$ Following such a definition the continuity ...
0
votes
2answers
30 views
Is this piecewise function continuous at the origin?
$f(x,y)$ is defined to be $\frac{x}{|y|}\sqrt{x^2+y^2}$ when $y \neq 0$ and $0$ when $y=0$. Is $f(x,y)$ continuous at $(x,y)=(0,0)$?
I don't know why, but I can't seem to find two paths that yield ...
2
votes
2answers
40 views
Holder condition for $x^\beta$
Let $f(x)=x^\beta$ (for some fixed $0<\beta<1$) be defined on $(0,1)$. It's not hard to see that $f$ is $\beta$-Holder.
How can I prove that $x^\beta$ is not $\alpha$-Holder for ...
0
votes
0answers
16 views
Unique continuos linear function given a continuous function from a dense space in X to Y (Y is a Banach Space).
Let $X$ be a normed space, let $Y$ be a Banach Space, let $D\subseteq X$ be a dense linear subspace of $X$ and let $L:D\rightarrow Y$ be a continuous linear function. Then there is a unique continuous ...
0
votes
1answer
25 views
Continuous function involved with integrals and limit
Let $f:[0,\infty)\rightarrow R$ be a continuous function such that for all $A>0$ the integral $\int_{A}^{\infty}\frac{f(t)}{t}dt$ converges. Suppose that $0<a<b$.
Show that
a. ...
1
vote
2answers
29 views
Homeomorphism $id_M:(M,\tau_d)\rightarrow(M,\tau_h)$
I am reading thorugh some topological definitions, in my book it is stated that $id_M:(M,\tau_d)\rightarrow(M,\tau_h),x\rightarrow x$ is a Homeomorphism where
$(M,d)$ is a metric space, ...
0
votes
3answers
34 views
preimage definition of continuity
I'm currently studying functional analysis and the professor covered continuity using the definition that the preimage of every open set is open. I can follow the definition, which basically means ...
1
vote
2answers
58 views
Doubt in proof of continuity
I was reading some examples of proving continuity using $\epsilon$-$\delta$ argument and well, I've found one that I'm not understanding one step. The problem is: prove that the function $f: ...
0
votes
2answers
43 views
Limit of series - exponential series
Series nd continuous functions
Question :
For $0<x<\infty ,$ let $$f(x) = \sum_{0}^{\infty } e^{-nx}$$ .Show that f(x) is continuous function.
Work Done:
I know every concept of sequences and ...
0
votes
3answers
48 views
Finding continuous f where f is not bounded .
I am finding a function f which is continuous on a closed interval but not bounded on the interval.
2
votes
1answer
51 views
Is $f(x,y)$ continuous?
I want to find out if this function is continuous:
$$(x,y)\mapsto \begin{cases}\frac{y\sin(x)}{(x-\pi)^2+y^2}&\text{for $(x,y)\not = (\pi, 0)$}\\0&\text{for $(x,y)=(\pi,0)$}\end{cases}$$
My ...
