Let be a sequence of functions from to defined by
Then which one of the following statements is true?

- Both the sequences and converge uniformly on .
- Neither nor converges uniformly on .
- converges pointwise but not uniformly on any interval containing the origin
- converges pointwise but not uniformly on any interval containing the origin.

TIFR 2019 GS Part A, Problem 12

Analysis

Hard

Real analysis Bartle, Sherbert

Do you really need a hint? Try it first!

Check the function is uniformly continuous or not using sup-norm definition i.e then the function is uniformly continuous if as

so as then function is uniformly continuous

Now, as in . Consider . Calculate . What can you say about option 4?

.

Can you comment about ?

It is clear that $latex f_n'(x)$ is not uniformly convergent on any such $ latex I$. So option 4) is correct

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