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2 C 20645
0
Evaluate ர்‌ (>

1
dx
What is Cauchy principle value. Find the principal value of | a
1

State Leibniz rule for differentiation of Ramann integrals.
State that ['(p+1)=p|p for p> 0.
(10 x 3 = 30 marks)
Section B

Answer at least five questions.

Each question carries 6 marks.

All questions can be attended.
Overall Ceiling 30.

Show that the Dirichlet’s function :

6 if x is rational

0 if xis irrational is not continuous at any point of R.

State and prove Bolzano intermediate value theorem.

Show that the following functions are not uniformly continuous on the given sets :
(8) f(x)= x? on A= [0,4].
(0) g(x)= शाप 1 ण) 8 = (0,0).

If f :[a,b]—R is continuous on [a,b], then show that f ¢R[a,b].

Let (ಗ) be a sequence of continuous functions on a set ACR and suppose that (/,) converges

uniformly on A to a function f: A— R. Then show that fis continuous on A.

Let f,, :[0,1] >IR be defined for n>2 by:

nx ,0 << छ
7
={-n® (x-2/n), Y. 2
ಗೈ (೫) 71-7೫ (ಜ-2/7), ಗೈ 5% 5
0 ‏کر‎ 71 < < 1.
1 1
Show that the limit function is Riemann integrable. Verify whether lim [7 n(X) = [7 (x)dx ,
0 0

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