Let be a sequence of functions from to . where . Then which of the following statement is true:

- and converge uniformly on .
- converges uniformly on but does not.
- converges uniformly on but does not.
- converges uniformly to a differentiable function on .

TIFR 2019 GS Part A, Problem 15

Analysis

Moderate

Real Analysis, Bartle and Sherbert

Do you really need a hint? Try it first!

Here we will check the sup norm condition. See the hints in question 12 of GS 2019 in cheenta portal and try this question again.

Observe that and

then as . Can you rule out any of the condition?

Calculate and draw the conclusion.

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