## Let $A\subset R^{n}$ . Then $A$ is disconnected iff there exists a continuous and surjective functon $f:A\to${0,1} December 2

Let $A\subset R^{n}$ . Then $A$ is disconnected iff there exists a continuous and surjective function $f:A\to${0,1} How can I prove this? To prove $\rightarrow$, I know that if $A$ is disconnected, then there are two open, non empty and disjoint sets

## Calculating the derivative with limited info. December 2

G(x) := integral(f(t)) dt from x to x^2Calculate G'(x).I've made some progress by integrating by parts with f(t) = 1(f(t)) but I'm stuck now and don't know where to go.

## Cyclic integration by parts trick December 2    2

I have seen cyclic integration by parts trick used for trignometric integrals. I got the impression that it is necessary for the trick to work that derivative of the function is cyclic. However it is not so, because the trick works for polynomials as

## Definite Integral involving reciprocals of logs December 2

Integrate $\int_2^{4e} \frac{1}{xln(x+1)}\,dx$I have tried partial fractions, u substitution and parts but i cant get the final answer out. my main problem is dealing with the $x$ and $x+1$ simultaneously. Integrating by partial fractions, i am left

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