sqt

How do I construct a homeomorphism between [0, 1) and [0, ∞)? I can prove they are homeomorphic because F(d, x) = dx is a homeomorphism [0, 1) → [0, d) for every d > 0, but I cannot find a specific function for d = ∞.

>>8476440
Use a circle.
> see wildbergers rational parameterization of a circle
or
> pughs analysis book, I think chapter 1

how to prove lim (x->0) (x+1)^3. I know that the answer should be L = 1 since f is continuous, but we can only use the limit definition. I also know we must choose an epsilon > 0, and then use |f(x) - L| = |(x+1)^3 - 1| to terms of |x-0| = |x|, but i don't see how except maybe expanding

>>8476571
Can you use continuity? You can write out the ε-δ definition for the continuity of the function from which the limit definition in terms of ε-δ follows immediately. If not, bound the entire function or simply bound x+1. You can use what you know about the limit in proving it.

>>8476455
Will look into this, thanks. I cant shake the feeling a more obvious function should exist, though.

>>8476440
>>8476598

doesn't the tangent function work?
like tan(pi*x/2)
as x approaches 1, tan(pi*x/2) approaches infinity
or did i misunderstand something

>>8476595
Thank you. Unfortunately we cannot use continuity, only the definition of a limit at a point.

>>8476440
x/(1-x)

>>8476571
want to show |(x+1)^3-1|< epsilon given |x|<delta.

so |(x+1)^3-1|=|x^3+3x^2+3x|<= |x|^3+3|x|^2+3|x| < delta^3+3delta^2+3delta

this last polynomial in delta is surjective since its leading power is odd, so you can make it smaller than epsilon

>>8476605
>or did i misunderstand something
You didnt, thanks!

>>8476652
I tried something like this fml thanks too!

I'm trying to develope a general strategy to tell whether a function is continuous and differentiable at a point.

First, if I can show that the function is not continuous at a point then i immediately know it isn't differentiable.

If our point of interest is a singleton we know f is continuous at that point (vacuously).

To my knowledge the only way to show a function is continuous at a point is toshow it satisfies the definition of continuity. If it is a piece wise function, and we are interested in a point at which f is defined differently on either side, we must examine the limit from both sides.

To show it is not continuous, i can negate the definition of continuity and show that that holds (which is sort of a pain in the ass i've found). Or, another popular method is to find two sequences in the domain that both converge to the point, but whose corresponding sequences of functional values converge to different values, or not at all.

Another way is to find one sequence of points in the domain that converge to the point of interest, but whose corresponding sequence of functional evaluations converge to somethign other than what the function evaluates to at that point.

I know differentiability implies continuity, so if I can show a function is differentiable I have shown it is continuous.

If the function is continuous at the point, then to show differentiability one must use the definition.

Is this all true? Sorry, I didn't think this would be so long.

How often should I do revision on past topics? I feel sometimes thag I'm losing problem solving skills even though I still have the concepts pretty clear.

>>8477167
You're trying to develop Cauchy Riemann equations again ?

How the fuck do I work out the right ascension and declination of a planet on the celestial sphere at a given time past some epoch when a measurement was last made? Suppose I know all the orbital elements of the planet

>>8477375
You must have me confused with another anon.

>>8477380
Its tough bro, I'll give you that.