- probability theory - Why does a C. D. F need to be right-continuous . . .
This fact is useful to resolve this natural question: Let $\{X_i\}_{i=1}^{\infty}$ be i i d random variables uniform over $[-1,1]$
- is bounded linear operator necessarily continuous?
Added @Dimitris's answer prompted me to mention, beyond the fact that the implication on normed spaces indeed is an equivalence, that it's the converse which holds in the wider context of topological vector spaces, while the proposition mentioned here fails: there are bounded discontinuous linear operators, yet every continuous operator remains
- functional analysis - continuity in the strong topology implies . . .
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- Difference between continuity and uniform continuity
I understand the geometric differences between continuity and uniform continuity, but I don't quite see how the differences between those two are apparent from their definitions For example, my book
- Proving the inverse of a continuous function is also continuous
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- Why Do We Care About Hölder Continuity?
The regularity of Brownian paths is expressed quite precisely by it being $\alpha$-Holder continuous for all $\alpha< \frac 12$ but not $\alpha$-Holder continuous for $\alpha \geq \frac 12$ In fact, this generalizes to a lot of solutions of various SPDEs, where the solution ends up having Holder-regular sample paths $\endgroup$
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