Real Analysis Answers

If f is continuous on [a, b] and integral a to b of f(x)g(x)dx=0 for all continuous functions g on [a, b] then f is identically equal to 0 on [a, b]

For each of the following relations, indicate whether they are reflexive, symmetric, and transitive (for example select reflexive and transitive if the relation is reflexive and transitive, but not symmetric).

(a) Let R be the relation on N given by xRy if and only if x divides y.
(b) Let X be a set and let R be the relation "⊆" on . In other words, SRT if and only if S⊆T.

If y=-2x^(3)-3x, determine dy/dt when x=1 and dx/dt=4

dy/dt=?

For each of the following relations, indicate whether they are reflexive, symmetric, and transitive (for example select reflexive and transitive if the relation is reflexive and transitive, but not symmetric). (a) Let R be the relation on N given by xRy if and only if x divides y. (b) Let X be a set and let R be the relation "⊆" on . In other words, SRT if and only if S⊆T. (c) Let S be the set of people in the world. Define R on S by xRy if and only if x likes y (use common sense) (d) Let R be the relation {(1,1), (1,2), (2,2),(1,3),(3,3)} on the set {1,2,3}.

For each of the following relations, indicate whether they are reflexive, symmetric, and transitive (for example select reflexive and transitive if the relation is reflexive and transitive, but not symmetric).

(a) Let R be the relation on N given by xRy if and only if x divides y.

(b) Let X be a set and let R be the relation "⊆" on . In other words, SRT if and only if S⊆T.

(c) Let S be the set of people in the world. Define R on S by xRy if and only if x likes y (use common sense)

(d) Let R be the relation {(1,1), (1,2), (2,2),(1,3),(3,3)} on the set {1,2,3}.

(e) Let R be the relation on R given by xRy if and only if x−y∈Q.

(f) Let R be the relation on R given by xRy if and only if x−y∉Q.

(g) Let R be the relation on R given by xRy if and only if (x−y)2<0.

(h) Let R be the the relation on R given by xRy if and only if |x-y|≤2

(i) Let R be the relation {(1,1), (1,2), (1,3), (2,1), (2,2),(3,1)} on the set {1,2,3}.

(j) Let R be the relation on R given by xRy if and only if xy∈Q.

Compute lower and upper integrals o0f the function
f(x)
=4, 1<=x<=2
=3, 2<x<=4
=2, 4<x<=5

Is this function Riemann integrable on [1,5]? Justify.

Uniform continuity:

Prove that if f and g are uniformly continuous on R and are bound, then f*g is uniformly continuous on R.

if f(x)=x and g(x)=(1/x), find (f/g)(x)

how to prove this equation A∩B⊂A⊂A⊂B

What are the points of continuity and the points of discontinuity of the following function? Justify your answer.
f(x) = x if x is rational.
f(x) = x^3 if x is irrational.