Question

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 continuous on R

Answer #1

A function f : A −→ R is uniformly continuous and its domain A ⊂
R is bounded. Prove that f is a bounded function. Can this
conclusion hold if we replace the "uniform continuity" by just
"continuity"?

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

Consider the function f : R → R defined by f(x) = ( 5 + sin x if
x < 0, x + cos x + 4 if x ≥ 0. Show that the function f is
differentiable for all x ∈ R. Compute the derivative f' . Show that
f ' is continuous at x = 0. Show that f ' is not differentiable at
x = 0. (In this question you may assume that all polynomial and
trigonometric...

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].

2. Suppose [a, b] is a closed bounded interval. If f : [a, b] →
R is a continuous function, then prove f has an absolute minimum on
[a, b].

Prove that the function f : R \ {−1} → R defined by f(x) = (1−x)
/(1+x) is uniformly continuous on (0, ∞) but not uniformly
continuous on (−1, 1).

If f is a continuous, positive function defined on the interval
(0, 1] such that limx→0+ = ∞ we have seen how to make sense of the
area of the infinite region bounded by the graph of f, the x-axis
and the vertical lines x = 0 and x = 1 with the definition of the
improper integral.
Consider the function f(x) = x sin(1/x) defined on (0, 1] and
note that f is not defined at 0.
• Would...

Prove that the function f(x) = x2 is uniformly
continuous on the interval (0,1).

Show that the function f(x)=x2sin(x) is uniformly
continuous on [0,b] for any constant b>0, but that is not
uniformly continuous on [0,infinity)

Use each definition of a continuous function to prove that every
function f: Z --> R is
continuous

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