# Tagged Questions

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) ...

40 views

### Given a continuous function with an asymptote, prove that the function is uniformly continuous.

I state the exercise: Given $f: [0, + \infty) \rightarrow R$, f continuous. Prove that if $\lim_{x \rightarrow \infty} f(x) = \lambda$, where $\lambda \in R$ then $f$ is uniformly continuous. My ...
51 views

35 views

### Homeomorphisms between any two doubly punctured spheres and two punctured $R^n$.

Let $p, q$ be the north pole and the south pole of $S^n$ respectively. Then $S^n-p-q$ is homeomorphic to $S^n-a-b$ where $a,b$, are distinct points in $S^n$. Also $R^n-a$ is homeomorphic to $R^n-b$...
37 views

### Example of a continuous surjection from $I=[a,b], a,b \in \mathbb{R}$ to $S^2$.

Continuous surjection from $I=[a,b], a,b \in \mathbb{R}$ to $S^2$. What is an example of such a surjection? I can't think of any. I would greatly appreciate any examples.
23 views

### find all values of a and b for which the function will be continuous

I have to find all values of a and b for which the function will be continuous. what I do is next: ...
31 views

### Non zero continuous path $[0,1]\to \mathbb C$ has continuous logarithm

Let $\gamma:[0,1]\to \mathbb C$ be continuous, and not passing through $0$. How can we prove that, using complex analysis, there is a continuous $G:[0,1]\to \mathbb C$ so that $\gamma=e^G$ ? This can ...
37 views

8 views

51 views

### When is ${{x^2y} \over {(x^2+y^2)^\alpha}}$ continuous, using polar-coordinates

Given $$f ({x,y})= \begin{cases} {{x^2y} \over {(x^2+y^2)^\alpha}},&(x,y) \ne {(0,0)}\\ 0,&(x,y)={(0,0)} \end{cases}$$ For what values of $\alpha$, $f$ is continuous in ${(0,0)}$? I set ...
57 views

32 views

47 views

### The space of continuous fuctions is compact - Other direction!

we all know that if $X$ is a compact topological space then $F(X)$ is compact for all $F\colon X\to \mathbb{R}$ continuous. I was wondering whether the converse is true? For metric spaces I have found ...
40 views

### Finding a transformation that yields a prescribed PDF

I am attempted to procure a function from a composition when given the PDF (I typed the full problem at the bottom in its entirety in case I left out details in my inquiry). I understand how to get ...
### Show that there exists a partition $-\infty=t_0<t_1<…<t_k=\infty$ such that $\lim_{t\rightarrow t_j^{-}} F(t)-F(t_{j-1})<\epsilon$
Consider a real-valued random variables $X$ defined on the probability space $(\Omega, \mathcal{F}, \mathbb{P})$ with cumulative distribution function $F(t):=\mathbb{P}(X\leq t)$. I want to show that ...