Some Talks by Peter Jipsen

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A sequence of continuous functions that converges to a discontinuous function:


A sequence of differentiable functions that converge to a discontinuous function:


A sequence of functions for which the limit of the integrals is not equal to the integral of the limit:


And again a differentiable sequence with the same property (on the interval `[0,1]`):




`=-n e^-n-n e^(-nx)/n]_0^1=-n e^-n-e^-n+1=1-(n+1)/e^n->1` and `n->oo`.


However, the pointwise limit of `n^2xe^(-nx)` is zero, so the integral of this limit is not `1`.


(Interesting note: while creating this example I found an error in "Counterexamples in Analysis" where the sequence is given as `nxe^(-nx)`.

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