Problem 4. Assume F is of bounded variation and continuous. Prove that F = F1 −...

Problem 4. Assume F is of bounded variation and continuous. Prove that F = F₁ − F₂, where both F₁ and F₂ are monotonic and continuous.

Expert Solution

Colby Messinger answered 2 years ago

Prove that if f is a bounded function on a bounded interval [a,b] and f is...

Prove that if f is a bounded function on a bounded interval [a,b] and f is continuous except at finitely many points in [a,b], then f is integrable on [a,b]. Hint: Use interval additivity, and an induction argument on the number of discontinuities.

Let f : R → R be a function. (a) Prove that f is continuous on...

Let f : R → R be a function. (a) Prove that f is continuous on R if and only if, for every open set U ⊆ R, the preimage f −1 (U) = {x ∈ R : f(x) ∈ U} is open. (b) Use part (a) to prove that if f is continuous on R, its zero set Z(f) = {x ∈ R : f(x) = 0} is closed.

Prove or provide a counterexample If f is T_C−T_U continuous, then f is T_U−T_C continuous. Where...

Prove or provide a counterexample If f is T_C−T_U continuous, then f is T_U−T_C continuous. Where T_C is the open half-line topology and T_U is the usual topology.

a) Suppose f:R → R is differentiable on R. Prove that if f ' is bounded...

a) Suppose f:R → R is differentiable on R. Prove that if f ' is bounded on R then f is uniformly continuous on R. b) Show that g(x) = (sin(x4))/(1 + x2) is uniformly continuous on R. c) Show that the derivative g'(x) is not bounded on R.

Suppose f maps a closed and bounded set D to the reals is continuous. Then the...

Suppose f maps a closed and bounded set D to the reals is continuous. Then the Uniform Continuity Theorem says f is uniformly continuous on D. To prove this, we will suppose it is not true, and arrive at a contradiction. So, suppose f is not uniformly continuous on D. If f is not uniformly continuous on D, then there exists epsilon greater than zero and sequences (a_n) and (b_n) in D for which |a_n -b_n| < 1/n and |f(a_n)...

1. Prove that if a set A is bounded, then A-bar is also bounded. 2. Prove...

1. Prove that if a set A is bounded, then A-bar is also bounded. 2. Prove that if A is a bounded set, then A-bar is compact.

9. Let f be continuous on [a, b]. Prove that F(x) := sup f([x, b]) is...

9. Let f be continuous on [a, b]. Prove that F(x) := sup f([x, b]) is continuous on [a, b]

1) The graph of the derivative f ' of a continuous function f is shown. (Assume...

1) The graph of the derivative f ' of a continuous function f is shown. (Assume the function f is defined only for 0 < x < ∞.) (a) On what interval(s) is f increasing? (2a) On what interval(s) is f decreasing? (b) At what value(s) of x does f have a local maximum? (2b) At what value(s) of x does f have a local minimum? x=? (c) On what interval(s) is f concave upward? (2c) On what interval(s) is...

4. Let a < b and f be monotone on [a, b]. Prove that f is...

4. Let a < b and f be monotone on [a, b]. Prove that f is Riemann integrable on [a, b].

True and False (No need to solve). 1. Every bounded continuous function is integrable. 2. f(x)=|x|...

True and False (No need to solve). 1. Every bounded continuous function is integrable. 2. f(x)=|x| is not integrable in [-1, 1] because the function f is not differentiable at x=0. 3. You can always use a bisection algorithm to find a root of a continuous function. 4. Bisection algorithm is based on the fact that If f is a continuous function and f(x1) and f(x2) have opposite signs, then the function f has a root in the interval (x1,...

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