7
votes
Accepted
Can we make this subspace $\aleph_0$-dimensional?
I claim that your space is the same as $\mathcal C(\bar A, \Bbb R)$. In particular the restriction map $f \mapsto f|_A$ from $\mathcal C(\bar A, \Bbb R)$ to your space is an isomorphism.
The ...
- 3,886
4
votes
Accepted
Number of continuous points of a graph (Problem 34 from 97-99 Math GRE practice questions booklet)
We furnish an example for which there are exactly two points of continuity, and no more:
$$f(x) = \begin{cases} \sqrt{1 - x^2}, & x \in [-1,1] \cap x \in \mathbb Q \\
-\sqrt{1 - x^2}, & x \in [...
- 137k
3
votes
Accepted
Show that the jump $j_f(c)$ of increasing $f$ at $c$ is given by $\inf\{f(y)-f (x): x < c < y, x, y \in I\}$.
It is enough to show the following.
If $A$ and $B$ are subsets of $\mathbb R$ such that $\inf(A)$ and $\sup(B)$ make sense, then $\inf(A-B)=\inf(A)-\sup(B)$, where $A-B=\{a-b\mid a\in A,b\in B\}$.
...
- 4,574
3
votes
Accepted
Topological continuity of a circle parametrization
The preimage of an open set by a continuous map must be open in the domain. In this case, $[0,1)$ has the topology induced by $\mathbb{R}$, that is, open sets in $[0,1)$ are open sets of $\mathbb{R}$ ...
- 4,391
3
votes
Discontinuous solution of $y''(x)-2(1-x)(y'(x))^2=0$ with $y(0)=1$ and $y(2)=-1$
Solutions of equations always depend on the solution concept you choose, that is, you first have to define what you mean by a solution. However, your problem is a classical Dirichlet boundary value ...
- 7,099
3
votes
Is open map needed to be continuous?
Consider the spaces $X = \mathbb{R}$ with the standard topology, and $Y = \mathbb{R}$ with the discrete topology. Define the map $f: X \rightarrow Y$ by $f(x) = x$, which is the identity map.
$f$ is ...
- 365
2
votes
Accepted
limit of convergent series in point wise convergent series of continuous functions
Your example is good. Here is another example with each $f_n$ also being smooth:
Consider $f_n : [0, 1] \to \mathbb{R}$ given by $f_n (x) = n x \exp(-n x)$. Note that $f_n (0) = 0$ for all $n \in \...
- 7,255
2
votes
Accepted
Discountinous function has a zero.
That's actually an interesting problem. One way is repeating one of the proofs of the IVT. Denote $a_1=a, b_1=b$, and let $h_1=\frac{a_1+b_1}{2}$. If $f(h_1)=0$ then we are done. If $f(h_1)>0$ then ...
- 40.2k
2
votes
Accepted
Continuity and Product Spaces
The function $h$ is continuous if and only if $f,g$ are.
For one direction, assume $f,g$ are continuous and pick a net $(x_\alpha,y_\alpha)_\alpha$ in $A\times C$ converging to $(x,y)$. Then $\lim_\...
- 2,057
2
votes
Accepted
Prove that if $\lim_{x\to c}f'(x)$ exists, the value is $\lim_{x\to c} {f(x)-f(c)\over x-c}$ when $f$ is continuous.
You have proved the result already but you are just overthinking.
Suppose $\lim_{x\to c}f'(x)=l$. Then, given $\epsilon >0$ there exists $\delta >0$ such that $|f'(x)-l|<\epsilon$ whenever $0 ...
- 37k
2
votes
Accepted
If $f$ is Lipschitz and $g \in C^\infty_c(\mathbb{R})$, is $g \circ f$ Lipschitz?
If $f \in C^1_c(\mathbb{R})$, then $f^\prime$ is continuous with compact support, so $|f^\prime| $ attains its maximum, $L_f$, say. By the mean value theorem, for given $x, y$ there is $\xi\in (x,y)$ ...
- 22.4k
2
votes
Accepted
"No infinite discrete subspace" vs "No infinite pairwise disjoint family of opens"
To summarize and amend comments:
For $R_1$ (in particular, Hausdorff) spaces, the conditions are equivalent. But examples for $A \not \Rightarrow B$ exist among spaces which satisfy $T_1$, or weaker ...
- 26.2k
2
votes
Accepted
If $f$ is continuous at $a$, is $f^{-1}$ continuous at $f(a)$?
A counterexample is as follows (now $I$ is open). Define $f:(-1,+\infty)\to\mathbb{R}$ given by:
$$f(x) =
\begin{cases}
x & \text{ $x\neq \dfrac{1}{n}, x\neq n$} \\
\dfrac{1}{2n} &\text{$x=\...
- 4,391
2
votes
If $f$ is continuous at $a$, is $f^{-1}$ continuous at $f(a)$?
If you relax the condition that $I$ is open, then I believe this is a counterexample.
Let $f:[0,+\infty) $ be given by $$f(x) =
\begin{cases}
x & \text{$x \in \mathbb{Q}$} \\
-x & \text{$x \...
- 4,391
2
votes
Is the function $(x^2+y^2)\sin{(\frac{1}{x^2+y^2})}$ differentiable at the point $(0,0)$?
Since you already showed that $\frac{\partial{f}}{\partial{x}}$ and $\frac{\partial{f}}{\partial{y}}$ both exist at the point $(0,0)$ and are equal to $0$, your theorem 1 indicates that if $f$ is ...
- 35.3k
2
votes
Is $f(x,y)=\frac{3x^2-5y^2\sin(x)}{|x|+|y|}$ continous at $(0,0)$?
$\frac {3x^{2}} {|x|+|y|}\leq \frac {3x^{2}} {|x|}=3|x| \to 0$ and $\frac {|5y^{2}\sin x|} {|x|+|y|} \le \frac {5y^{2}} {|y|} =5|y| \to 0$. So $f(x,y) \to 0$ as $(x,y) \to (0,0)$. $f$ is not defined ...
- 37k
2
votes
Prove or disprove $C:=\{x\in\mathbb{R}^n|f(x)\leq f(x_0)\}$ is closed
Take $n=2$ and define $f$ as $0$ in the open unit disk and $1$ outside.
Then, if $x_0=(0,0)$, $C$ is the open unit disk, which is not closed.
- 4,391
2
votes
Prove or disprove $C:=\{x\in\mathbb{R}^n|f(x)\leq f(x_0)\}$ is closed
If $f$ is not continuous, this is clearly false. You can basically choose the values in any way you want. For example define $f:\mathbb{R}\to\mathbb{R}$ by $f(x)=0$ for $x\in(-1,1)$, and $f(x)=1$ ...
- 40.2k
1
vote
Accepted
Continuity of a mapping on a space of polynomial
General theoretical arguments were given in comments. More explicitely, a rough but sufficient bound is obtained via evaluation maps (as suggested in @julio_es_sui_glace's comment), for instance $$u_{...
- 35.3k
1
vote
A question about continuous functions with compact support in locally compact topological groups.
Funny, I was reading this part of the book just a few days ago, and it took me hours until I figured it out. I wish some authors would bother to write a few more details.
Let $K$ be any compact set ...
- 40.2k
1
vote
Subset of $C^1([0 , 1])$ is open
Suppose $f \in G$. Since $f'$ is continuous we see that $\epsilon=\inf \{|f'(x)|: 0 \le x \le 1\}>0$. If $\|f-g\|<\epsilon$ then
$g'(x)\neq 0$ for all $x$ because $|f'(x)-g'(x)|<\epsilon$ for ...
- 37k
1
vote
Finding the discontinuities of $\text{inf}_i (|x - q_i| 2^i)$
Given any sequnce $(\epsilon_n)_{n\in\mathbb{N}}$ with $\epsilon_n>0$ and $\sum_{n\geq 0}\epsilon_n<\infty$ one can look at the set $\mathbb{R}\setminus\bigcup_{n\geq 0}(q_n-\epsilon_n, q_n+\...
- 84
1
vote
Accepted
Is this Result on Continuity of Composite functions true?
No. Take $f = g = \chi_\mathbb{Q}$, the indicator function of $\mathbb{Q} \subset \mathbb{R}$, i.e.
$$
f(x) = g(x) = \begin{cases}
1 & \text{if } x \in \mathbb{Q} \\
0 & \text{if ...
- 2,892
1
vote
Accepted
Continuity of Probability for Events with Density Greater Than a Threshold
Since $\lim_{y \to b+} \mathbb P(p(X) \ge y) = \mathbb P(p(X) > b)$ and
$\lim_{y \to b-} \mathbb P(p(X) \ge y) = \mathbb P(p(X) \ge b)$, the only issue
is whether there can be $b$ such that $\...
- 450k
1
vote
Discontinuous solution of $y''(x)-2(1-x)(y'(x))^2=0$ with $y(0)=1$ and $y(2)=-1$
$\phi = y'$
$$\begin{align}\phi'-2(1-x)\phi^2=0&\implies\phi'=2(1-x)\phi^2\\&\implies\phi^{-2}\frac{d\phi}{dx} = 2(1-x)\\&\implies -\phi^{-1} = 2x-x^2+C_1\\&\implies \phi(x) = {\frac{1}...
- 608
1
vote
Accepted
If $f(0)=0,f(1)=1$,find all $a$ such that $\exists \xi\in (0,1)$ such that $f(\xi)+a=f'(\xi)$
This is problem 7 from last year's IMC (2023), as can be found here.
I know, because I composed this problem :]
For these kinds of problems, you often want to use Rolle's Theorem, which states that if ...
- 1,879
1
vote
If $f(0)=0,f(1)=1$,find all $a$ such that $\exists \xi\in (0,1)$ such that $f(\xi)+a=f'(\xi)$
We want to reduce set of different possibilities for $a$, so it makes sense to try to find a function s.t. $f'(x) - f(x)$ has low number of possible values. If it has just one, then $f'(x) - f(x) = a$,...
- 15.5k
1
vote
Differentiability of a Dirichlet Function Modified with $x^2$
$$0<\frac{|g(x)|}{|x|} \leq |x|$$
$$\lim_{x \to 0} 0 \leq \lim_{x \to 0}\frac{|g(x)|}{|x|} \leq \lim_{x \to 0} |x| \implies \lim_{x \to 0} |\frac{g(x)}{x}| = 0 \iff \lim_{x \to 0} \frac{g(x)}{x} = ...
- 1,508
1
vote
Accepted
Regularity and B-S equation
As mentioned in https://en.wikipedia.org/wiki/Geometric_Brownian_motion we have
$$S_{t}=S_{0}\exp \left(\left(\mu -{\frac {\sigma ^{2}}{2}}\right)t+\sigma W_{t}\right).$$
So the mapping $(\mu, \sigma)\...
- 3,611
1
vote
Find a continuous function from Moore plane to $\mathbb{R}$
It seems that you already know that the space is completely regular. So all you need to prove is that $X$ is closed in it. Because then it is a simple matter of applying the definition of being ...
- 43.1k
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