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.

Expert Solution

Colby Messinger answered 2 years ago

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.

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.

Let f be a one-to-one function from A into b with B countable. Prove that A...

Let f be a one-to-one function from A into b with B countable. Prove that A is countable. Section on Cardinality

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.

5). Let f : [a,b] to R be bounded and f(x) > a > 0, for...

5). Let f : [a,b] to R be bounded and f(x) > a > 0, for all x in [a,b]. Show that if f is Riemann integrable on [a,b] then 1/f : [a,b] to R, (1/f) (x) = 1/f(x) is also Riemann integrable on [a,b].

prove or disppprove. Suppose A & B are sets. (1) A function f has an inverse...

prove or disppprove. Suppose A & B are sets. (1) A function f has an inverse iff f is a bijection. (2) An injective function f:A->A is surjective. (3) The composition of bijections is a bijection.

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].

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.

f : [a, b] → R is continuous and in the open interval (a,b) differentiable. f...

f : [a, b] → R is continuous and in the open interval (a,b) differentiable. f rises strictly monotonously ⇒ ∀x ∈ (a, b) : f ′(x) > 0. (TRUE or FALSE?) f rises strictly monotonously ⇐ ∀x ∈ (a, b) : f ′(x) > 0. (TRUE or FALSE?) f is constant ⇐⇒ ∀x∈(a,b): f′(x)=0 (TRUE or FALSE?) If f is reversable, f has no critical point. (TRUE or FALSE?) If a is a “minimizer” of f, then f ′(a)...

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]

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