Question

Let f: [a,b] to R be continuous and strictly increasing on (a,b). Show that f is strictly increasing on [a,b].

Answer #1

Let f, g : [a, b] ---> R continuous such that
(f(a) - g(a)) (f(b) - g(b)) < 0.
a) Show that sup{|f(x) - g(x)| : x ϵ [a, b]} is strictly
positive and
achieved (is a maximum).

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...

Prove that: If f : R → R is strictly increasing, then f is
injective.

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

Let 0 < a < b < ∞. Let f : [a, ∞) → R continuous R at
[a, b] and f decreasing on [b, ∞). Prove that f is bounded
above.

Let f, g : [a, b] ---> R continuous such that
(f(a) - g(a)) (f(b) - g(b)) < 0.
b) Show that inf {|f(x) - g(x)| : x ϵ [a,b]} = 0 and is achieved
(is
a minimum).

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].

Let f : [a, b] → R be bounded, and assume that f is continuous
on [a, b). Prove that f is integrable on [a, b].

6. Let a < b and let f : [a, b] → R be continuous. (a) Prove
that if there exists an x0 ∈ [a, b] for which f(x0) 6= 0, then Z b
a |f(x)|dxL > 0. (b) Use (a) to conclude that if Z b a |f(x)|dx
= 0, then f(x) := 0 for all x ∈ [a, b].

Let F be continuous, show that f([a,b]) is a closed
interval.

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