Tagged Questions
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, ...
5
votes
2answers
103 views
If $f$ and $g$ are continuous, prove $f\circ g$ is continuous.
Suppose that $(X,T)$, $(Y,U)$ and $(Z,V)$ are three topological spaces and that
$g\colon X\to Y$ and $h\colon Y \to Z$ are continuous. Prove that $h\circ g\colon X \to Z$ is a continuous ...
5
votes
3answers
72 views
Continuous function that take irrationals to rationals and vice-versa. [duplicate]
Can someone help me?
How can I prove that there isn't an everywhere continuous function $f:\mathbb R \rightarrow \mathbb R$ that transforms every rational into an irrational and vice-versa?
0
votes
4answers
113 views
Find a function $f:\Bbb R \to \Bbb R$ which is discontinuous at $1,\frac 12,\frac 13, … $ but is continuous at every other point
(a) Find a function $f:\Bbb R \to \Bbb R$ which is discontinuous at $1,\frac 12,\frac 13, ... $ but is continuous at every other point.
(b) Find a function $f:\Bbb R \to \Bbb R$ which is ...
3
votes
5answers
121 views
Sequence of continuous functions which converges to a continuous limit
Any help with this: construct a sequence of continuous functions defined on $ [0,1] $ which converges pointwise but not uniformly to a continuous limit ?
Thank you.
2
votes
1answer
25 views
Differentiability of first derivative of a function
If a function $f$ is differentiable on domain $D$ and $f'$ is increasing on $D$, is $f'$ necessarily continuous on $D$? Is $f'$ necessarily differentiable on $D$? Counterexamples?
From Darboux ...
3
votes
1answer
58 views
Cauchy's functional equation - a generalisation? (do additive maps have to be continuous?)
If a map $f : \mathbb{R} \to \mathbb{R}$ is additive, in the sense that $f(x + y) = f(x) + f(y)$, then it is simple to show that $f$ is $\mathbb{Q}$-linear, buy it does not need to be ...
11
votes
4answers
311 views
$f(16x)=16f(x) $ and $ f$ is continuous
$f: \mathbb{R}\rightarrow \mathbb{R}$ is a continuous function such that $f(16x)=16f(x)$ for every real $x$.
Should it be $f(x)=ax$? How can I prove that?
9
votes
3answers
207 views
Solve the functional equation $f(x)=f\left({x\over 3}\right)+f\left({2x\over 3}\right)$ with $f : [0,\infty) \to \mathbb R$ continuous
Solve the functional equation
$$f(x)=f\left({x\over 3}\right)+f\left({2x\over 3}\right)\qquad \forall x\geq 0$$
with $f : [0,\infty) \to \mathbb R$ continuous.
I can't manage to get this one ...
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 ...
0
votes
3answers
57 views
question about continuity
The question is: Assume $f$ is a bounded continuous function in $[a,b)$ and differentiable in $(a,b)$. Also assume $f$ is not continuous in $[a,b]$. prove: $f'(x)$ is neither upper bounded nor lower ...
0
votes
0answers
28 views
Is Rayman function continuous and another question about the subject
The question as was written in title is whether the Rayman function is continuous where the Rayman function is this one: http://i.stack.imgur.com/xgFVy.png
and another question (if I can): if I have ...
2
votes
0answers
23 views
Monotonic function non-continuous in each rational [duplicate]
How can I prove that exists a monotonic non-decreasing function $f: [0,1] \rightarrow \mathbb R$ that isn't continuous in every rational of its domain?
0
votes
3answers
50 views
Construct function with 2 local minima at x1 and x2
I am trying to construct a continuous differentiable function $f(x)$ that for $x_1$ and $x_2$ takes the value $0$ and have global minimum at these points, i.e. $f(x_1)=f(x_2)=0$ and ...
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 ...
0
votes
1answer
56 views
Continuity of a function, Differentiable function
The following function is given:
$$f:\mathbb{R}\rightarrow \mathbb{R}, \ x\rightarrow \begin{cases} x^2\cos{\left(\frac{1}{x}\right)} & \text{for } x \neq 0\\ 0& \text{for } x =0\end{cases}$$
...
2
votes
1answer
171 views
Proof that a continuous function is bounded below
I have this question:
Assuming the theorem that a continuous real-valued function on a
closed bounded interval is bounded and attains its bounds, prove that
if $f\colon\mathbb R\to\mathbb R$ ...
1
vote
1answer
99 views
Extension of continuous function
The question is: Let $(K,\rho)$ be compact metric space. $F\subset K$ closed. $f:F\rightarrow \mathbb{R}$ continuous. Is there a continuous extension of $f$ on $K$?
Attempt: Suppose there exists ...
2
votes
0answers
53 views
Extreme Value Theorem and Semicontinuity
Restricting us to function of a single real variable, I was used to prove Extreme value theorem via the short way: show that continuous functions preserve compactness, and the job is done.
Now, I ...
3
votes
1answer
125 views
Please explain this example in Spivak
From page 105 of the 1994 edition of Spivak's Calculus:
A continuous function is sometimes described, intuitively, as one
whose graph can be drawn without lifting your pencil from the paper.
...
4
votes
0answers
77 views
Function that is discontinuous only for integer fractions
I have this question:
Find a function $f :\mathbb R \to\mathbb R$ which is discontinuous at the points of the
set $\{\frac1n : n \text{ a positive integer}\} \cup \{0\}$ but is continuous ...
0
votes
1answer
163 views
How to prove this limit composition theorem?
If $$\displaystyle \lim_{x \rightarrow c}f(x)=l$$ and $$\displaystyle
\lim_{x \rightarrow l}g(x)=L$$ and $f(x) \neq l$ in some punctured
neighbourhood of c,
then $\displaystyle \lim_{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 $0$ ...
0
votes
1answer
118 views
Show that there exist positive constants $a$, $b$ such that $|f(x)| \leq a |x| + b$ for every $x \in \mathbb{R}$.
Problem
For a uniformly continuous function $f:\mathbb{R} \rightarrow \mathbb{R}$, show that there exist positive constants $a$, $b$ such that $|f(x)| \leq a |x| + b$ for every $x \in \mathbb{R}$.
...
0
votes
0answers
37 views
What are the features of $n/d, n \rightarrow d, d \rightarrow \infty; n, d \in \mathbb{N} $?
What are the features of $n/d, n \rightarrow d, d \rightarrow \infty; n, d \in \mathbb{N} $?
What is the value of $\lim_{n \rightarrow d, d\rightarrow \infty} (n/d)$? What is the function's range? ...
1
vote
3answers
99 views
If $f$ is continuous on $\mathbb R$, is $f$ also continuous on $\mathbb R-\{0\}$?
I mean, if I make the latter claim, am I precluding the possibility that $f$ could actually be continuous on the entire reals? (I am right now proving the continuity of a 2-piece function joined in a ...
0
votes
2answers
158 views
Composing two discontinuous functions into a continuous one
Please help me think of an example of two discontinuous functions on $\mathbb R$ whose composition gives a continuous function on $\mathbb R$.
1
vote
1answer
98 views
Continuity of a function with 2 variables
Im stuck in this exercise:
Study the continuity of the next function:
$$f(x,y) = \begin{cases} \frac{x\sin(x^2+y^2)}{x^2+y^2}&\text{si } (x,y)\not=(0,0)\\ 0 &\text{si }(x,y) =(0,0). ...
0
votes
1answer
61 views
Steps to follow to find the continuity of function with 2 variables
I'm studying for my exam and I have a bit of trouble with these kind of exercices, since I have no theory and im a bit lost:
Study the continuity in (0,0) (g(x,y) is a function depending of the ...
4
votes
3answers
215 views
$\epsilon$-$\delta$ proof that $f$ is continuous for $x\notin\mathbb Q$ but isn't for $x\in\mathbb Q$
I'm trying to give an $\epsilon$-$\delta$ proof that the following function $f$ is continuous for $x\notin\mathbb Q$ but isn't for $x\in\mathbb Q$.
Let $f:\mathbb{A\subset R\to R}, \mathbb{A=\{x\in ...
4
votes
4answers
166 views
Prove that $f$ is not continuous for any value in $\mathbb R$.
I'm having some trouble in the following proof:
Let $f:\mathbb{R\to R}$ be given by:
$$
f(x) = \begin{cases}
1,&x\in\mathbb Q
\\
0,&x\notin\mathbb Q
\end{cases}
$$
Prove that $f$ is not ...
5
votes
2answers
93 views
$\epsilon$-$\delta$ proof that $f(x) = x \sin(1/x)$, $x \ne 0$, is continuous
I'm doing an exercise that asks me to prove that $f$ is continuous using a $\epsilon$-$\delta$ proof. I have that
$$
f(x) = \begin{cases}
x\cdot \sin \frac1x,&x\neq 0
\\
0,&x = 0
\end{cases}
...
1
vote
2answers
135 views
prove that $f(x)=\log(1+x^2)$ is Uniform continuous with $\epsilon ,\delta$ …
I have to prove that $f(x)=\log(1+x^2)$ is Uniform continuous in $[0,\infty)$ (with $\epsilon ,\delta$ formulas...)
I wrote the definition: (what I have to prove):
$\forall \epsilon>0 \quad ...
6
votes
1answer
401 views
Show f is uniformly continuous on $(a,b)$ if it is continuous and $\lim\limits_{x\to a^+}f(x)$ and $\lim\limits_{x\to b^-}f(x)$ exist
Let $f:(a,b)\to\mathbb{R}$ be continuous at all $x\in(a,b)$. If $\lim\limits_{x\to b^-}f(x)$ and $\lim\limits_{x\to a^+}f(x)$ exist in $\mathbb R$, how can we prove that $f$ is uniformly continuous on ...
7
votes
4answers
970 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, $$|f(x_1)-f(x_2)|=|\sin x_1 - \sin x_2|\le|\sin x_1|+|\sin ...
0
votes
1answer
36 views
choice of $\lambda$ when negating pointwise continuity
EDIT: To make sense, the function in the question must be some $f:[0,1]\times [0,1]\rightarrow \mathbb{R}$. Then, what I want to show is that $f$ is continuous with respect to its second argument. ...
4
votes
1answer
113 views
prove $x \mapsto x^2$ is continuous
I am to show the continuity of this function with the help of $\epsilon$-$\delta$ argument.
The function is: $g: \Bbb{R} \rightarrow \Bbb{R}$, $x \mapsto x^2$.
Given the $\epsilon$-$\delta$ ...
1
vote
1answer
52 views
check for Continuity $x \rightarrow 2x^4-2$
Given the function: $f:\Bbb{R}\rightarrow \Bbb{R},\quad x \rightarrow 2x^4-2$
i am asked to check for Continuity for all values of $x$. i am now overasked how to do this since $\Bbb{R}$ is not a ...
7
votes
5answers
367 views
Study continuity of this function
Hello im studying calculus at the university and I dont know how to solve the following exercise:
Study the continuity of the next function:
$$f(x,y) = \begin{cases} \frac{x^2-xy}{x+y}&\text{for } ...
3
votes
3answers
166 views
Coming up with an example, a function that is continuous but not uniformly continuous
What would be a example of a function that is continuous, but not uniformly continous? will f(x)=1/x be a example? Give domain from (0,2), why? Strictly, in definitions
1
vote
1answer
75 views
Showing the function $\chi$ is uniformly continuous
Let $(A,\rho)$ be a compact metric space and let $f: A \to A$ be a function satisfying $$ \forall \quad x,y\in A, \ x \ne y \ \ \implies \rho(f(x),f(y))< \rho(x,y). $$
Now define the function ...
0
votes
1answer
97 views
Let $f$ be a real valued sequentially continuous function relative to a closed bounded interval $I=[a,b]$. Prove that the set $f(I)$ is bounded above
The hint that I've been given is: for each n in the naturals, use the assumption that $n$ is not an upper bound for $f(I)$ to choose a sequence of $x_n$ (from $n=1$ to infinity) in $I$; then apply ...
6
votes
1answer
95 views
$C$ be a closed subset of the Cantor set $\Delta$. Show the existence of a continuous function $f:\Delta\to C$ s.t. $f(x)=x$, $x\in C$
Question: Let $C$ be a closed subset of the Cantor set $\Delta$. Prove there is a continuous function $f$ from $\Delta$ onto $C$ s.t. for every $x \in C$ we have $f(x)=x$.
Context: Advanced ...
3
votes
1answer
96 views
Very basic question about the definition of continuous of a functions
Suppose say $f:\{0,1\}\to \{1,2\}$ is $f$ continuous? Say $f(0)=2,f(1)=1$ I know the definition of continuous function. In my point of view, i think it is continuous as we can simply take ...
0
votes
1answer
119 views
Continuity of $\sqrt{x}$ at a point
I have some problem to understand the definition of a continuous function in a point.
I have $f(x) = \sqrt{x}$ and I want to check the continuity of the function above in the point $x_0 = 0$ or for ...
1
vote
2answers
82 views
Continuity at an arbitrary point
Let $f:\Bbb N \to\Bbb R$ by writing $f(n) = \frac1{n^2}$. Is $f$ continuous at any point in its domain.
So, my thought is, $f$ is a function with domain $\Bbb N$ - natural numbers, so each point ...
1
vote
2answers
143 views
continuity, discontinuity examples of functions
i. I need to find a function from R to R discontinuous at the even integers
5
votes
1answer
102 views
A question about uniform continuity
Let $F$ be a continuous function on the real set $\mathbb R$ such that the function $x \mapsto xF(x)$ is uniformly continuous on $\mathbb R$ . Prove that $F$ is also uniformly continuous on $\mathbb ...
0
votes
0answers
72 views
necessary and sufficient condition for continuous function to be monotonic at certain domain
Lately I found there exists a function which is "continuous but nowhere monotonic".
So, now I want to know that (the title).
I'm really thank you if you give me a proof of it.
1
vote
1answer
74 views
Continuity and pushing a limit inside the function's domain
Consider some right-continuous function $f:\mathbb{R\cup\{-\infty,\infty\}}\to [0,1]$. I have to evaluate (i) $\lim_{b \to 0^+} f(\frac{a}{b})$, and (ii) $\lim_{b \to 0^-} f(\frac{a}{b})$ where $a \in ...
