# Understand the problem

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 .

##### Source of the problem

TIFR 2019 GS Part A, Problem 15

##### Topic

Analysis

##### Difficulty Level

Moderate

##### Suggested Book

Real Analysis, Bartle and Sherbert

# Start with hints

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.

#### Can you recall any of the property of uniform convergence of sequence of functions which will call the conclusion?

#### If a sequence of function is uniform convergent then a continuous sequence of functions will converge to a continuous function.

Now the given sequence of function is continuous but the limit is not, hence this is not a uniform convergence. So, option c is correct.# Watch the video (Coming Soon)

# Connected Program at Cheenta

#### College Mathematics Program

The higher mathematics program caters to advanced college and university students. It is useful for I.S.I. M.Math Entrance, GRE Math Subject Test, TIFR Ph.D. Entrance, I.I.T. JAM. The program is problem driven. We work with candidates who have a deep love for mathematics. This program is also useful for adults continuing who wish to rediscover the world of mathematics.