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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3answers
1k views
How to show that $f(x)=x^2$ is continuous at $x=1$?
How to show that $f(x)=x^2$ is continuous at $x=1$?
4
votes
1answer
100 views
A problem on continuous functions
$f : S^1 \rightarrow \mathbb{R}$ is a continuous map. Define $$A = \{(x, y) \in S^1 \times S^1: x \neq y, f(x) = f(y)\}$$
We want to prove that A has uncountably many points. It seems very evident, ...
5
votes
1answer
243 views
Is a continuous function simply a connected function?
Intuitively, a function $\mathbb{R}\rightarrow\mathbb{R}$ is continuous if you can draw its graph without taking the pen off the page. This suggests the following theorem:
A map $f:X \rightarrow Y$ ...
11
votes
4answers
227 views
If $f:\mathbb{R}\to\mathbb{R}$ is continuous and $f(x)\neq x$ for all $x$, must it be true that $f(f(x))\neq x$ for all $x$?
Let $f: \Bbb R → \Bbb R$ be a continuous function such that $f(x)=x$ has no real solution . Then is it true that $f(f(x))=x$ also has no real solution ?
6
votes
1answer
313 views
Is there a monotonic function discontinuous over some dense set?
Problem (for fun--not homework)
Can we construct a monotonic function $f : \mathbb{R} \to \mathbb{R}$ such that there is a dense set in some interval $(a,b)$ for which $f$ is discontinuous at all ...
3
votes
1answer
190 views
Show that $f$ is uniformly continuous if limit exists
Let $f(x)$ be continuous on $(0,1]$. Show that $f$ is uniformly continuous IFF $\displaystyle \lim_{x\to0^+} f(x)$ exists.
Thoughts:
Backward Proof:
Let another function $\overline f(x)$ be ...
3
votes
2answers
174 views
Proving that cosine is uniformly continuous
This is what I've already done. Can't think of how to proceed further
...
3
votes
2answers
160 views
Continuous mappings pull back closed sets to closed sets
George F Simmons, Topology and Modern Analysis pg.79 Problem 4
Let $X$ and $Y$ be metric spaces. Show that an into mapping $f:X \rightarrow Y$ is continuous $\iff$ $f^{-1}\left(G\right)$ is closed in ...
1
vote
2answers
83 views
Find a general control and then show that this could have been achieved at x2
Determine the general form of $u_0, u_1 ~\text{and} ~ u_2$ if a system of difference equations of the form
$$x_{n+1} = Ax_n + Bu_n,$$
where:
$$A = \begin{pmatrix}
3 & 2 & 2 \\
-1 ...
0
votes
2answers
65 views
Continuous function, not sure what to do here…
The question is as follows:
Let $f(x) = \begin{cases} x, & \mbox{if } x<1 \\ x^2+1, & \mbox{if } x\ge 1 \end{cases}$
Let $g$ be a function such that $fg$ is continuous at $1$, and ...
14
votes
2answers
481 views
If $\lim_n f_n(x_n)=f(x)$ for every $x_n \to x$ then $f_n \to f$ uniformly on $[0,1]$?
This is a self-posed question, so I do not know the answer and I would like to know what do you think about.
Let $f,f_n:[0,1]\to \mathbb R$ be continuous functions. Assume that for every sequence ...
6
votes
1answer
179 views
pointwise limit on a complete metric space
Let $\{f_n: X\rightarrow \mathbb{R}\}$ be a sequence of continuous real-valued functions on a complete metric space, $X$. Suppose this sequence has a pointwise limit, $f$. How easy is it to see that ...
5
votes
2answers
118 views
$f$ brings convergent nets to convergent nets, is it continuous?
Let $f:(X,\mathcal T)\to (Y,\mathcal S)$ be a function between topological spaces. Let for any convergent net $(x_\alpha)$ in $X$, $(f(x_\alpha ))$ be convergent in $Y$. Is $f$ continuous?
(It seems ...
3
votes
2answers
365 views
Dini's Theorem. Uniform convergence and Bolzano Weierstrass.
In Spivak's chapter on uniform convergence he asks to prove the following
THEOREM Let $\{f_n\}$ be sequence of continuous functions that converge pointwise to $0$ over $[a,b]$. If $0\leq ...
2
votes
3answers
123 views
Uniform Continuity of $f(x)=x^3$
1.)Determine whether $f(x)=x^3$ is uniformly continuous on [0,2)
So far, I have $\delta$ = 2 and $\epsilon$ = 8, and plan on using the sandwich theorem with $x^2$ and eventually equating $\delta = ...
2
votes
5answers
663 views
limits and Continuity
I had my first encounter with Calculus a decade ago. Back then it was purely mechanical. Formulas and rules of derivation and integration were being written on the board without deriving it and were ...
1
vote
2answers
115 views
Left topological zero-divisors in Banach algebras.
Let $ A $ be a unital Banach algebra. Define $ \zeta: A \longrightarrow [0,\infty) $ by
$$
\forall a \in A: \quad \zeta(a) \stackrel{\text{def}}{=} \inf_{b \in \mathbb{S}(A)} \| ab \|,
$$
where $ ...
7
votes
4answers
972 views
Prove $\sin x$ is uniformly continuous on $\mathbb R$
How do I prove $\sin x$ is uniformly continuous on $\mathbb R$ with delta and epsilon?
I proved geometrically that $\sin x<x$ and thus, $$|f(x_1)-f(x_2)|=|\sin x_1 - \sin x_2|\le|\sin x_1|+|\sin ...
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 ...
3
votes
2answers
40 views
Find the points of discontinuity: $f(x) = (x^4+x^3+2x^2)/\tan^{-1}(x)$ if $x\ne0$ and $f(0)=10$
Here is the question:
Find the points of discontinuity:
$$f(x) =
\begin{cases}
\frac{x^4+x^3+2x^2}{\tan^{-1}x} & \text{if} \ x\ne0 \\ \\
10 & \text{if} \ x=0 \\
\end{cases}.$$
...
2
votes
2answers
34 views
Determine the value of $a$ for which the function is continuous at $x=0$
Determine the value of $a$ for which the function is continuous at $x=0$
$$
f(x) =
\begin{cases}
a \sin\frac{\pi}{2}(x+1) & \text{if} \ x\le0, \\ \\
\frac{\tan x - \sin x}{x^3} ...
1
vote
1answer
95 views
Hard Limit proof!
Suppose $(a_n)$ and $(b_n)$ are sequences where $b_n$ is increasing and approaching positive infinity. Assume that $\lim_{n\to \infty}$ $\frac{a_{n+1}-a_n}{b_{n+1}-b_n}=L$, where $L$ is a real number. ...
1
vote
3answers
478 views
Show $f$ is constant if $|f(x)-f(y)|\leq (x-y)^2$.
Problem: Let $f$ be defined for all real $x$, and suppose that
$$|f(x)-f(y)|\le (x-y)^2$$
for all real $x$ and $y$. Prove $f$ is constant.
Source: W. Rudin, Principles of Mathematical Analysis, ...
0
votes
1answer
56 views
Continuity of a function, Differentiable function
The following function is given:
$$f:\mathbb{R}\rightarrow \mathbb{R}, \ x\rightarrow \begin{cases} x^2\cos{\left(\frac{1}{x}\right)} & \text{for } x \neq 0\\ 0& \text{for } x =0\end{cases}$$
...
6
votes
2answers
514 views
Show that $f'$ is not continuous at 0 for the following function:
$$ f(x) = \begin{cases} x + 2x^2\sin(1/x) & \text{ for }x \neq 0 \\ 0 & \text{ for } x = 0\end{cases} $$
This is another exam practice question I am working on.
I simply took the ...
4
votes
2answers
108 views
Continuity, Real Analysis
Some T/F questions. Instead of doing strict proofy questions, I am trying to understand the topic and making sure whether I am clear on the topic. Let me know whether I am right or wrong and I'll ...
3
votes
1answer
55 views
How to show that a continuous map on a compact metric space must fix some non-empty set.
Suppose $(X,d)$ is a compact metric space and $f:X\to X$ a continuous map. Show that $f (A)=A$ for some nonempty $A\subseteq X.$
I start this by supposing that $A_0:=X$ and $A_{n+1}:=f(A_n)$ for ...
3
votes
1answer
114 views
A continuous bijective map between two path-connected topological spaces, that is not a homeomorphism?
I am trying to think an example of a continuous bijective map between two path-connected topological spaces, that is not a homeomorphism.
I am looking for an example in $\mathbb{R^n}$. Any help ...
3
votes
0answers
74 views
Prove that if $f$ is uniformly continuous then the one sided limit $\lim_{x\to 0^+} f(x)$ exists. [duplicate]
If $f(x)$ is a continuous function on $(0,1]$, prove that if $f$ is uniformly continuous, then the one sided limit $\lim_{x\to 0^+} f(x)$ exists.
3
votes
5answers
113 views
Let $\displaystyle f$ be a continuous function from $[0,4]$ to $[3,9].$
I came across the following problem that says:
Let $\displaystyle f$ be a continuous function from $[0,4]$ to $[3,9].$ The which of the following options is correct?
$1.$ there must be an $x$ ...
3
votes
1answer
120 views
$f$ is continuous at $c$ $\implies$ $f$ has a limit at $c$. True?
Further to Another simple/conceptual limit question where I was questioning David Brannan's assertion in his A First Course in Mathematical Analysis that $f(x)=\sqrt x,x\geq 0$ has no limit at $0$ ...
3
votes
1answer
120 views
Continuity understanding the definition and images and preimages
I am having trouble understanding exactly the difference between the epsilon delta definition for continuity and the one for the limit of a function.
epsilon greater than 0, there exists a δ such ...
3
votes
1answer
322 views
Is the total variation function uniform continuous or continuous?
I have been doing some excercises on total variation when the following questions came up to my mind:
(1) Let $f$ be continuous on the interval $[0,1]$ and be of bounded variation. Is it true that ...
2
votes
1answer
42 views
Continuity of $\max$ of Lebesgue integral
Let $m$ be a probability measure on $Z \subseteq \mathbb{R}^p$, so that $m(Z)=1$.
Consider a locally bounded $f: X \times Y \times Z \rightarrow \mathbb{R}_{\geq 0}$, with $X \subseteq \mathbb{R}^n$, ...
2
votes
1answer
29 views
Baire one functions, characteristic functions of intervals
Do you think you could help me prove that characteristic functions of intervals are Baire one functions?
And is it true that linear combinations of Baire one functions are also Baire one?
2
votes
2answers
41 views
How to prove that $|||y |||$ is continuous using the usual basis of $\Bbb R^{n}$
How to prove that $|||y |||$ is continuous on $\Bbb R^{n}$ by using the usual basis of $\Bbb R^{n}$
By the way, $||| \cdot |||$ is a norm on $\Bbb R^{n}$
I can show this by using triangle ...
2
votes
1answer
115 views
Continuous function proof by definition
Prove that if $f$ is defined for $x\ge 0$ by $f(x)=\sqrt x$, then $f$
is continuous at every point of its domain.
Definition of a continuous function is:
Let $A\subseteq\mathbb{R}$ and let ...
2
votes
1answer
93 views
How do I apply an epsilon delta proof to the following problem?
Any help in solving the following problem would be greatly appreciated:
Let $f, g_1, g_2$ be functions from $\mathbb R$ to $\mathbb R$, with
$g_1(x) \leq f(x) \leq g_2(x)$, for all $x \in ...
2
votes
2answers
96 views
Continuity of $f \cdot g$ and $f/g$ on standard topology.
Let $f, g: X \rightarrow \mathbb{R}$ be continuous functions, where ($X, \tau$) is a topological space and $\mathbb{R}$ is given the standard topology.
a)Show that the function $f \cdot g : X ...
2
votes
3answers
613 views
Does $\lim_{h\rightarrow 0}\ [f(x+h)-f(x-h)]=0$ imply that $f$ is continuous?
Suppose $f$ is a real function defined on $\mathbb{R}$ which satisfies
$$\lim_{h\rightarrow 0}\ [f(x+h)-f(x-h)]=0.$$
Does this imply that $f$ is continuous?
Source: W. Rudin, Principles of ...
2
votes
1answer
243 views
What can we say about functions satisfying $f(a + b) = f(a)f(b) $ for all $a,b\in \mathbb{R}$? [duplicate]
Possible Duplicate:
Is there a name for such kind of function?
I am investigating functions satisfying the exponentiation identity $f(a + b) = f(a)f(b)$ for all $a,b\in \Bbb R$. This is ...
2
votes
2answers
293 views
Proof of Bolzano's Theorem
I know one proof of Bolzano's Theorem, which can be sketched as follows:
Set
$f$ a continuous function in $[a,b]$ such that ${f(a)<0<f(b)}$.
${A=\{x:a<x<b \text{ and } f <0\in[a,x] ...
1
vote
4answers
87 views
we need to show it is discontinuous at x≠0
can any one just explain to me to me answer
Q)
$$f(x)=\begin{cases}
x &\text{if }x\in \mathbb{Q} \\
0 &\text{if }x\in \mathbb{R}\setminus\mathbb{Q}
\end{cases}$$
we need to show it is ...
1
vote
4answers
46 views
Let $f$ be continuous on the real numbers. Let $c$ be in real number with $f(x)=c$ for all $x$ in $\mathbb{Q}$. Show that $f(x)=c$ on $\mathbb{R}$.
Can someone solve this question?
Let $f$ be continuous on $\mathbb{R}$. Let $c$ be in real number with $f(x)=c$ for all $x$ in $\mathbb{Q}$.
Show that $f(x)=c$ for all $x$ in $\mathbb{R}$.
...
1
vote
0answers
97 views
A question regarding uniform continuity
Let $f, g:[0,1] \to [0, \infty]$ be continuous. Assume $f(x)>g(x)$ for all $x \in [0,1]$. Prove that there exists a $M>1$ such that $f(x) \ge M g(x)$ for all $x \in [0,1]$.
1
vote
2answers
109 views
Derivative based on continuity
I have a question about whether I am even close to correct.
Let $\mathbb{I}$ and $\mathbb{J}$ be open intervals, and the functions $f:\mathbb{I} \to R$ and $h:\mathbb{J}\to R$ have the property that ...
1
vote
3answers
182 views
Is the function $f( x)=1/|x|^{1/2}$ Lipschitz continuous?
Is the function $f( x)=1/|x|^{1/2}$ Lipschitz continuous near $0$? If yes, find a constant for some interval containing $0$
I think the answer is yes since I can find $L=1$ that satisfies Lipschitz ...
1
vote
3answers
168 views
Fixed point in a continuous function
Suppose that $f$ is a function defined in $[a;b]$ to $[a;b]$ and continuous on $[a;b]$.
The problem is I haven't the definition of the function, this is more abstract, but even if how can I prove that ...
0
votes
1answer
46 views
Logical Relations Between Three Statements about Continuous Functions
(a) $f$ is continuous almost everywhere
(b) there exists a continuous function $g$ such that $f = g$ almost everywhere (on every set of non-zero measure)
(c) $f$ is nearly a ...
0
votes
2answers
74 views
A question on a continuous function
Let $f: X\rightarrow Y $, and $Y$ is Hausdorff compact. Show that
$f$ is continuous iff the graph $G_f=\{(x, f(x): x \in X\}$ is closed in $X \times Y$.
Thanks ahead.
