Let F n(x) =1/n *e^(-n^2*x^2) if x element of R ,n=1,2 .....
Prove that fn tend to 0 uniformly on R ,that derivative of fn tend to 0 point wise on R ,but that the convergence of {fn'} is not uniform on any interval containing the origin
Here
And
so
Now attains maximum value
tending to 0 as
Let us take the interval [a,b] containg 0.
Thus
Which does not tends to zero as
Hence the sequence is not uniformly convergent in any interval [a,b] containing the origin.
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