Yeah, it's harder than it looks

Replace $f$ by a step function (with infinitely many steps), then find $g$ in this case.

Guess: Use a bump function that is $0$ on $\left[0,t_0\right]$ and then bounded above by the constant value of $f$ on $\left[t_0,\infty\right]$.

I think if you get $g(x) = \inf_{t \geq x}\left\{f(t)\right\}$ you can get what you want, except for the strictly increasing

Oh wait, I missed the strictly increasing part, but it should work if we make it linear on $\left[0,t_o\right]$ in a way such that it just touches the lowest non-zero local minimum of $f$

@Thorgott This doesn't works, since we need ...

Yeah, that is it ^^

@user8469759 If we exclude the strictly increasing condition, we could take $g=f$

@Karl Your idea is interesting

Use the function $\hat f(x)=\min_{t\ge 1/n}f(t)$ for $\frac 1{n+1}<x\le\frac 1n$. Then, let $g$ be the line segments between the points $\left(\frac 1{n+1},\hat f\left(\frac 1{n+1}\right\right)$ and $\left(\frac 1n,\hat f\left(\frac 1n\right)\right)$. Only problem is if $\hat f$ does not change on the interval $(\frac 1{n+2},\frac 1n]$ but that is easily repaired.

The part where $f$ becomes constant needs adjustment too.

Something like using $h=\frac 12\cdot g$ for the portion before $x=t_0$, then using a strictly increasing bounded function for the rest will handle the latter issue.

Oh, nice

This seems to work

Thanks

Oh, my latex is wrong above.

The first point is $\left(\frac 1{n+1},\hat f\left(\frac 1{n+1}\right)\right)$

I'm too lazy to run mathjax :)

@Derso No, because your function $f$ might oscillate.

anyone have any good resources for propositional + first order logic for people who are new to it and want a comprehensive curriculum?

Bourbaki - Theorie des Ensembles

@user8469759 That's why I said "if we exclude the strictly increasing condition".

upsetting all logicians in the room

@Derso what I meant instead is excluding "strictly"

but still increasing

hilbert-ackermann proof calculus with epsilon (tau) terms what a boss

monotone

@user8469759 monotone increasinf

@user8469759 Oh, I see. You're right

Hello guys! I am wondering when I should use - or + in the complex Descartes theorem. Maybe I could find the solution myself by a lot of trial and error but I feel like I already wasted enough time... Any insights?

anyone knows good video lectures on analysis that uses Rudin book ?

Hello @TedShifrin! How are you? Do you remember our last discussion about an exercise?

How could I possibly forget?

Hey @TedShifrin :)

Hi Perturb

You know I was looking through G&P again earlier today, and only now I appreciate what it does after seeing the abstract stuff in Lee's book

I definitely recommend learning stuff from G&P before doing the graduate abstract stuff.

And there's beautiful material on transversality and intersection numbers that aren't in most graduate courses.

Thanks for the advice, but I'm going to have to do the abstract stuff starting in a weeks time anyway, but I'm not too worried about it, but I'll definitely using G&P for inspiration when I get lost in the abstract stuff

When I taught the graduate diff geo course, after doing all the tensors and Stokes's Theorem stuff, I always taught a unit on curves and surfaces using the fancier techniques (moving frames and differential forms) before doing general Riemannian geometry.

That sounds really cool, working with concrete stuff is always good as I've learned :p

@TedShifrin Have you been following the tennis recently?

hey hey hey