Real Analysis Answers

Let f be of bounded variation on [a,b] and define V is,

V(x)= {0 if x=a

V₁(a,x) if a less than or equal to x.less than or equal to b

Then show that a point of continuity of f is also a point of continuity of V and conversely

Show that a function of bounded variation on [a,b] is bounded therein

Let alpha be of bounded variation on [a,b] and assume that f element of R(alpha) on [a,b] .show that f element of R(alpha) on every subinterval [c,d] of [a,b]

Assume f element of R(alpha) on [a,b] where alpha is of bounded variation on [a,b] .let V(x) denote the total variation of alpha on [a,x] if a less than x less than or equal to b and let V(a) =0 .show that | integral a to b fdalpha| less than or equal to integral a to b |f| dv less than or equal to MV (b).

Suppose alpha increases on [a,b] a less than or equal to x₀ less than or equal to b ,alpha is continuous at x₀ ,f(x₀) =1 and f(x )=0 if x not equal to x₀ .prove that f element. Of R(alpha) and that integral a to b fdalpha =0

Let f_n(x) =1/(1+n²x²) if 0 less than or equal to x less than or equal to 1,n=1,2,........ Prove that {f_n} converges point wise ,but not uniformly on [0,1] Is term by term integration permissible?

Assume that. F_n converges to f uniformly on S if each f_n is continuous at a point c of S .show that the limit function f us also continuous at c

Let F n(x) =1/n *e^(-n^2*x^2) if x element of R ,n=1,2 .....

Prove that fn tend to 0 uniformly on R ,that derivative of fn tend to 0 point wise on R ,but that the convergence of {fn'} is not uniform on any interval containing the origin

Show that the sequence {𝑓𝑛} where 𝑓𝑛(𝑥) = 𝑛𝑥(1 − 𝑥)

𝑛 does not converge uniformly on [0, 1].

Check whether the following sequences (sn) are Cauchy, where (i)sn=1+2+3+...+n 4n3 + 3n (ii) sn 3n + n2