Question

Let f: R --> R be a differentiable function such that f' is bounded. Show that f is uniformly continuous.

Answer #1

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

Let f:Ω-->R^m be uniformly continuous on Ω⊂R^n. Show if (Ω)
is bounded, then f(Ω) is bounded.

Let f : R → R be a continuous function which is periodic. Show
that f is bounded and has at least one fixed point.

We know that any continuous function f : [a, b] → R is uniformly
continuous on the finite closed interval [a, b]. (i) What is the
definition of f being uniformly continuous on its domain? (This
definition is meaningful for functions f : J → R defined on any
interval J ⊂ R.) (ii) Given a differentiable function f : R → R,
prove that if the derivative f ′ is a bounded function on R, then f
is uniformly...

show that if f is a bounded increasing continuous function on
(a,b), then f is uniformly continuous. Hint: Extend the function to
[a,b].

4a). Let g be continuous at x = 0. Show that f(x) = xg(x) is
differentiable at x = 0 and f'(0) = g(0).
4b). Let f : (a,b) to R and p in (a,b). You may assume that f is
differentiable on (a,b) and f ' is continuous at p. Show that f'(p)
> 0 then there is delta > 0, such that f is strictly
increasing on D(p,delta). Conclude that on D(p,delta) the function
f has a differentiable...

Show that the series \sum_{n=1}^{\infty} 1/(x^2 + n^2) defines a
differentiable function f: R -> R for which f' is
continuous.
I'm thinking about using Cauchy Criterion to solve it, but I got
stuck at trying to find the N such that the sequence of the partial
sum from m+1 to n is bounded by epsilon

Let f be a bounded measurable function on E. Show that there are
sequences of simple
functions on E, {(pn) and {cn}, such that {(pn} is increasing and
{cn} is decreasing and each of
these sequences converges to f uniformly on E.

Let f : R → R be a function satisfying |f(x) − f(y)| ≤ 3|x −
y|^{1/2} for all x, y ∈ R. Apply E − δ definition to show that f is
uniformly continuous in R.

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.

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