Is there a way to prove that if $f:\left[ a,b \right]\to\mathbb{R}$ is a continuous function and $f(a) = f(b)$, then there exists a $\epsilon > 0$ such that $f$ takes the value $c = f(a) + \epsilon$ exactly $2$ times in $(a,b)$?
or, somewhat more elaborately, nonconstant functions that achieve maxima on [a,b] at a and b but have a minimum less than f(a) = f(b) somewhere in the middle.
i'm reluctant to think too hard because if i find something new, i fear the hypotheses will change. good luck. it seems like a counterexample will be 'subtle' but i don't see any obvious reason why there couldn't be one.
@Prithubiswasleftmse yes, but with adding more and more troughs could give you a counterexample. I have not tried adding more troughs but I think it's possible. Add troughs such that for every epsilon>0 the height of the trough is <epsilon.
Based on what I said earlier. Consider f(0)=f(1)=0 and f(1/n)=1/n, for every n>2. And define f(n*)=1/2, where n* is the middle point of 1/n and 1/(n+1). Connect the points (n*, 1/2) and (1/n,1/n) by a straight line and similarly connect (1/(n+1),1/(n+1)) to (n*,1/2) by a straight line.
Call this a graph of f, which should give you a counterexample. @Prithubiswasleftmse
@Koro Prithu is correct; your function is discontinuous at 0, so it fails to be a counter-example. There is a counter-example, but yours isn't. @Prithubiswasleftmse @leslietownes: In fact, there is a continuous function from [a,b]→ℝ such that f(a) = f(b) = 0 < f↾(a,b) and #( { x : x∈[a,b] ∧ f(x) = v } ) ∈ {0,3,6} for every v > 0.
Inspired by the answers given by these three questions (here, here, and here), what is the general solution for constructing a group with a specific number of Sylow subgroups?
That is, given a prime $p$ and a positive integer $n\equiv1\pmod p$, is it always possible to construct a group $G$ with...
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ted, when munchkin announces an intention to carry the cat down a flight of stairs and picks the cat up, what would your intuition be? to intervene? or just see what happens?
Given an integral domain D, which is a Euclidean domain and given a,b in D find gcd(a,b). Am I supposed to first prove that D is an ED by finding that distance function that satisfies the properties that makes D an ED?
and with such distance discovered, then proceed using Euclidean algorithm?
I used the function that was identically $1$ for a while so that I would get a number of maxima so that there wasn't any place where a value was taken twice.
I bet! I got really into baking in days of yor. cookies, German chocalate cake, cream puffs, pies. My favorite of all was the basic shortbread cookie made with butter. The basic recipe is so versatile, for say, turtle cookies, or the batter mixed with ground pecans for pecan fingers. Now I'm getting hungry!!
If $X$ is an affine variety and $p \in X$ some point, then $\mathcal{O}_{X,p}$ denotes the local ring of an affine variety $X$ the point $p$. How exactly is this defined? I remember it being defined as equivalence classes of pairs $(f,U)$ where, I think, $f$ is a regular map on the open subset $U \subset X$ containing $p$...But I don't remember exactly what the equivalence relation is...
