Suppose /(x) = Va.x+b, where a > 0 and a, b are constants in R. What...
R such that f is integrable on every [a,b] (6) Suppose f is a function and a where b> a. Then we define the improper integral eb f(x)dx=lim | b-oo Ja f(x)da, if that limit exists. Assume that f(x) is continuous and monotonically decreasing on [0,00). Prove that Joof exists if and only if Σ f(n) converges. This result is known as the integral test for series convergence.
(12) Suppose that f: [0, o0) - (0, 00) and that f e R((0, n]), for every n E N. Prove that f is Lebesgue measurable, the Lebesgue integral Jo.0)f dA exists, and f dA [0,00) lim f (x)dx noo (12) Suppose that f: [0, o0) - (0, 00) and that f e R((0, n]), for every n E N. Prove that f is Lebesgue measurable, the Lebesgue integral Jo.0)f dA exists, and f dA [0,00) lim f (x)dx noo
(12) Suppose that f [0, oo) - [0, o0) and that f E R(0, n), for every n E N. Prove that f is Lebesgue measurable, the Lebesgue integral So.o)dexists, and f dA f (x)dax lim -- noo 0,00)
2. The domain of fug is given by: (a) R - (b)R-{0} (c)R-{-2,0)(d) R - {0} 3. The inverse of the function f(x) is a) ( )
3) (Assessment Topic: Continuity) Consider the function sgn(x)- (a) What is the domain of sgn(x)? (b) Prove that lim sgn(x)メ1? (Hint : Use a contradictory argument in conjunction with the E-δ method.) (c) Show that for any two positive constants p > 0 and r > 0, where 0<p<r, that lim, ++ee)0, by finding For fixed number q > 1 define the function for each e > 0. (d) Show that lim,-0+ K(t)0. (Hint: Write out a few terms on...
8. Let T: R+R be the function T(x) = mx +b, where m and b are some constants. Prove that T is a linear transformation if and only if b = 0.
(0, 1) given by f (x) - sin (). Is f Let f b e the function t on the domain uniformly continuous? Explain. (You may take it as given that sin is a continuous function) Suppose that f [0, oo) -R is a continuous function, and suppose also that lim, ->oo f (x)- 0. Prove that f is uniformly continuous Just to be clear: to say that lim,->o f (x) - 0 means that
8. Let T: R → R be the function T(x) = mx + b, where m and b are some constants. Prove that T is a linear transformation if and only if b = 0.
= mx + b, where m and b are some constants. Prove that T is a linear 8. Let T: R → R be the function T(x) transformation if and only if b = 0.
How to do this question? pleases show some steps. 6. Consider a sequence of continuous function fn : [a,b] → R. Suppose there exist constants 72 1 and ß > 0 independent or, p such that e b for any p > 1 and n e N. Show that there exists a constant C depending on independent of n, such that and B, but rb max lfn(x) rela,b for any n e N. [Hint: Results from Example 4.3.5 may be...