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]`):

`int_0^1n^2xe^(-nx)dx=n^2xe^(-nx)/(-n)]_0^1-int_0^1n^2e^(-nx)/(-n)dx`

 

`=-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)`.