C 20645 (Pages : 3) வி... ௮ ಸ...

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SIXTH SEMESTER U.G. DEGREE EXAMINATION, MARCH 2022
(CBCSS—UG)
Mathematics
MTS 6B 10—REAL ANALYSIS
(2019 Admissions)
Two Hours and a Half Maximum : 80 Marks
Section A

Answer at least ten questions.
Each question carries 3 marks.
All questions can be attended.
Overall Ceiling 30.
Define continuity of a function. Show that the constant function f (x)= bis continuous on R.

State Boundedness theorem. Is boundedness of the interval, a necessary condition in the theorem ?
Justify with an example.

If f: AIR is uniformly continuous on ACR and (x,,) is a Cauchy sequence in A. Then show

that f («,) is a Caychy sequence in R.

Define Riemann sum of a function / : [a,b] oR.

Suppose f and g are in R[a,b] then show that ¢ + & 18 77 [२ [५.९] .

State squeeze theorem for Riemann integrable functions.

If f belong to R[ a,b], then show that its absolute value |f | is in R[a,d].
Define pointwise convergence of a sequence (f,,) of functions.

Discuss the uniform convergence of 7, (2) -2” 07 (-1,1] .
If h,(x)=2nxe™ for xe[0,1],n¢Nand h(x)=0 for all x <[0,1], then show that :

1 1
lim fh, (x)dx # [h(x) dx.
0 0

State Cauchy criteria for uniform convergence series of functions.

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