# Tagged Questions

Intuitively, a continuous function is one where small changes of input result in correspondingly small changes of output. Use this tag for questions involving this concept. As there are many mathematical formalizations of continuity, please also use an appropriate subject tag such as (real-analysis) ...

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### $S$ be $\pi$-system on a set, given two measures on $\sigma(S)$, is there a topology on $\sigma(S)$ making $S$ dense, and the two measures continuous?

Let $\Omega$ be a non-empty set , $S \subseteq \mathcal P(\Omega)$ be a Pi system (https://en.wikipedia.org/wiki/Pi_system ) on $\Omega$ , let $\sigma(S)$ be the $\sigma$-algebra generated by $S$ (i.e....
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### How to show continuity of a function with $n-1$ exponentiations?

Say we are given a function $$\Gamma(x)=f_1(x)^{f_2(x)^{\cdot^{\cdot^{f_n(x)}}}}$$ where $f_i,i\in[1;n]$, are continuous functions in their domains. Also assume that the function makes sense, e.g., ...
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### How do you find one-sided limits *algebraically*?

Find $$\lim_{x\to\ -0.5^-}\sqrt{\frac{x+2}{x+1}}$$ Sorry, I have no idea where to start. I know how to find regular limits algebraically, but not one-sided. Thanks
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### If a function is continuous a.e., then it is measurable. [duplicate]

Is this true or wrong? How to prove it ？
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### Equivalence of pretopological continuity to customary continuity (a direct proof)

Can you provide a direct (not based on the neighborhood definition of pretopologies) proof for pretopological spaces (expressed as closure operators) that a function $f$ from a topological space $\mu$ ...
### Prove or Disprove that $f(x)$ cannot be $C_3$
Consider a function $y=f(x)$ with a single argument x with the real number line as its domain. Fix a real number $Q>0$, and suppose the following: $f(x)=f'(x)=f''(x)=0$ for all x from the interval ...
Let $f$ and $g$ be real-valued continuous functions on $\Bbb R^2$ that satisfy the following condition: $$x<y \implies x< f(x,y) < g(x,y) <y$$ Assume that there are two sequences \$\{a_n\...