MHB Is f Integrable and What Is the Limit of the Riemann Sum?

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We only have the epsilon-delta definition to work with for these.

Prove that f is integrable and verify the value. On [0,1] f(x)=1 if x=1/2 else 0. $$\int_{0}^{1} \,f$$ =0

Prove: If f is integrable on [0,1] then $$\lim_{{n}\to{\infty}}\ \frac{1}{n} \sum_{k=1}^{n} f(\frac{k}{n})$$ = $$\int_{0}^{1} \,f$$ .
 
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skippenmydesign said:
We only have the epsilon-delta definition to work with for these.

Prove that f is integrable and verify the value. On [0,1] f(x)=1 if x=1/2 else 0. $$\int_{0}^{1} \,f$$ =0

Prove: If f is integrable on [0,1] then $$\lim_{{n}\to{\infty}}\ \frac{1}{n} \sum_{k=1}^{n} f(\frac{k}{n})$$ = $$\int_{0}^{1} \,f$$ .

Wellcome on MHB skippenmydesign!...

... in general the definition of Riemann Integral is...

$\displaystyle \int _{a}^{b} f(x)\ dx = \lim_{\text{max} \Delta_{k} \rightarrow 0} \sum_{k=1}^{n} f(x_{k})\ \Delta_{k}\ (1)$

... where $x_{k}$ are aritrary points in the subintervals $\Delta_{k}$ of the interval [a, b] ... in Your case all the term of the sum (1) are 0 with only one exception for the subinterval containing $x = \frac{1}{2}$ where the term is $\Delta_{k}$, so that the limit is 0...

Kind regards

$\chi$ $\sigma$
 
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A sphere as topological manifold can be defined by gluing together the boundary of two disk. Basically one starts assigning each disk the subspace topology from ##\mathbb R^2## and then taking the quotient topology obtained by gluing their boundaries. Starting from the above definition of 2-sphere as topological manifold, shows that it is homeomorphic to the "embedded" sphere understood as subset of ##\mathbb R^3## in the subspace topology.
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