4:46 AM
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A real function is continous on $[a,b]$ such that either $f(a) > 0 >f(b)$ or $f(a) < 0 < f(b)$, then prove that $f(c) = 0$ for some $c \in (a,b)$. Proof :- We divide $[a,b]^*$ in $H$ equal parts, where $H$ is a positive infinite hyperinteger. We get, a, a +\delta, a+2\delta,\ ... \ , ...

This discussion about similar theorem would suggest that the tag can probably be removed:
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A search result for Mean Value Theorem gives us 2715 results, and results on the page are like ones I think we can include in the tag. The theorem is an important result in calculus, and questions relating to its applications, proofs. I think it would be useful if could have the tag, as it can gr...

I have removed the tag from the above question. If it is used again (in the same or in another question), I will make a post on meta. (Is seems that the user A---B created several tag in the last weeks.)
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Fix an Abelian category $\mathscr{A}$, and a homological spectral sequence $(E_{p,q}^r)$, so that $E^{r+1}\cong H(E^r)$, then, according to the ncatlab, we say that $E^r$ converges to $E^\infty$ if for all $(p,q)$, there exists some $r_0$ such that $r\ge r_0$ implies that $E^{r}_{p,q}\cong E^\inf... Monstrous Moonshine also created tag-excerpt for naturality. 1 hour later… 5:59 AM I have added here, since IMO it certainly belongs here. The question is specifically about subsets of metric space, but I do not see what could change in arbitrary topological spaces. So I was a bit hesitant whether to add also . Union of connected subsets is connected if intersection is nonempty 10 Let$\mathscr{F}$be a collection of connected subsets of a metric space$M$such that$\bigcap\mathscr{F}\ne\emptyset$. Prove that$\bigcup\mathscr{F}$is connected. If$\bigcup\mathscr{F}$is not connected, then it can be partitioned into two disjoint, non-empty subsets$A,B$. Let$x\$ be a...