Question

Let f : R → R be a bounded differentiable function. Prove that for all ε...

Let f : R → R be a bounded differentiable function. Prove that for all ε > 0 there exists c ∈ R such that |f′(c)| < ε.

Homework Answers

Know the answer?
Your Answer:

Post as a guest

Your Name:

What's your source?

Earn Coins

Coins can be redeemed for fabulous gifts.

Not the answer you're looking for?
Ask your own homework help question
Similar Questions
Let f: R --> R be a differentiable function such that f' is bounded. Show that...
Let f: R --> R be a differentiable function such that f' is bounded. Show that f is uniformly continuous.
Let a < b, a, b, ∈ R, and let f : [a, b] → R...
Let a < b, a, b, ∈ R, and let f : [a, b] → R be continuous such that f is twice differentiable on (a, b), meaning f is differentiable on (a, b), and f' is also differentiable on (a, b). Suppose further that there exists c ∈ (a, b) such that f(a) > f(c) and f(c) < f(b). prove that there exists x ∈ (a, b) such that f'(x)=0. then prove there exists z ∈ (a, b) such...
Let f be a function differentiable on R (all real numbers). Let y1 and y2 be...
Let f be a function differentiable on R (all real numbers). Let y1 and y2 be pair of numbers (y1 < y2) with the property f(y1) = y2 and f(y2) = y1. Show there exists a num where the value of f' is -1. Name all theroms that you use and explain each step.
Let I be an interval. Prove that if f is differentiable on I and if the...
Let I be an interval. Prove that if f is differentiable on I and if the derrivative f' be bounded on I then f uniformly continued on I!
a) Let f : [a, b] −→ R and g : [a, b] −→ R be...
a) Let f : [a, b] −→ R and g : [a, b] −→ R be differentiable. Then f and g differ by a constant if and only if f ' (x) = g ' (x) for all x ∈ [a, b]. b) For c > 0, prove that the following equation does not have two solutions. x3− 3x + c = 0, 0 < x < 1 c) Let f : [a, b] → R be a differentiable function...
Prove or give a counterexample: If f is continuous on R and differentiable on R∖{0} with...
Prove or give a counterexample: If f is continuous on R and differentiable on R∖{0} with limx→0 f′(x) = L, then f is differentiable on R.
Prove or give a counter example: If f is continuous on R and differentiable on R...
Prove or give a counter example: If f is continuous on R and differentiable on R ∖ { 0 } with lim x → 0 f ′ ( x ) = L , then f is differentiable on R .
let H be a subgroup of R. Assume ∃ ε ∈ R ε>0 such that (-ε...
let H be a subgroup of R. Assume ∃ ε ∈ R ε>0 such that (-ε ,ε) ∩ H= {0} Prove H cyclic
Let f : E → R be a differentiable function where E = [a,b] or E...
Let f : E → R be a differentiable function where E = [a,b] or E = (−∞,∞), show that if f′(x) not = 0 for all x ∈ E then f is one-to-one, i.e., there does not exist distinct points x1,x2 ∈ E such that f(x1) = f(x2). Deduce that f(x) = 0 for at most one x.
Suppose f is differentiable on a bounded interval (a,b) but f is unbounded there. Prove that...
Suppose f is differentiable on a bounded interval (a,b) but f is unbounded there. Prove that f' is also unbounded in (a,b). Is the converse true?