Sma2110:Real Analysis Question Paper
Sma2110:Real Analysis
Course:Bachelor Of Computer Science
Institution: Meru University Of Science And Technology question papers
Exam Year:2013
QUESTION ONE – (30 MARKS)
(a) Define a partition of a closed interval ??,?? . (2 Marks)
(b) Explain the statement ??2 in finer than ??1 (2 Marks)
(c) State the condition for a function f to be Riemann integrable over ??,?? . (1 Mark)
(d) Show that a constant function k is integrable and that b a abkdxk . (4 Marks)
(e) Prove that 0 , baV f iff ?? in a constant function on ??,?? . (4 Marks)
(f) Define pointwise convergence of a sequence of functions on a set E. (3 Marks) (g) Evaluate 23 232 1 lim 2 x x xx x (4 Marks)
(h) Show that the function defined by
0,1 0, 2sin
)(
xwhen xwhen
x
x xf has a removable
discontinuity at the origin. (5 Marks)
2
(i) Show that 2 1 2 11 dxf , where 1 3)( xxf (5 Marks)
QUESTION TWO – (20 MARKS)
(a) Prove that, if f is a function of bounded variation, then f is bounded. (10 Marks)
(b) Use the function defined by
0 0
,
,
0
2
cos)(
xif xif
x
xxf
to show that a bounded function need not be of bounded
variation. (10 Marks)
QUESTION THREE – (20 MARKS)
(a) Define uniform convergence of a sequence of functions ... ,3,2,1, nf n on a set E. (4 Marks) (b) Determine whether the function
4,0 4, 4 4 x x x x xf converges. (8 Marks) (c) Show that the function ?? ?? = ??2 is uniformly continuous on -1,1 . (8 Marks)
QUESTION FOUR – (20 MARKS)
(a) Prove that the series 1 ?????????? (i) is convergent if ?? > 1 (6 Marks) (ii) is divergent if ?? = 1 (8 Marks)
(b) Prove that if ) ( lim xf ax
exists, if must be unique. (6 Marks)
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