Question

Let f be a monotonic increasing function on a closed interval
[a, b]. Show that f is

Riemann integrable on [a, b].

Answer #1

For the given function f(x) = c show that it is
Riemann integrable on the interval [0, 1] and find the
Riemann integral

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

Proof: Let f and g be functions defined on (possibly different)
closed intervals, and assume the range of f is contained in the
domain of g so that the composition g ◦ f is properly defined If f
is integrable and g is increasing, then g ◦ f is integrable.

Let f: [0, 1] --> R be defined by f(x) := x. Show that f is
in Riemann integration interval [0, 1] and compute the integral
from 0 to 1 of the function f using both the definition of the
integral and Riemann (Darboux) sums.

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

(a) Show that the function f(x)=x^x is increasing on (e^(-1),
infinity)
(b) Let f(x) be as in part (a). If g is the inverse function to
f, i.e. f and g satisfy the relation x=g(y) if y=f(x). Find the
limit lim(y-->infinity) {g(y)ln(ln(y))} / ln(y). (Hint :
L'Hopital's rule)

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'(x)=x2-4x-5
Determine the interval(s) of x for which the function
is increasing, and the interval(s) for which the function is
decreasing.
Find the local extreme values of f(x) , specifying
whether each value is a local maximum value or a local minimum
value of f.
Graph a sketch of the graph with parts (a) and (b) labeled

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].

Show that Thomae's function is Darboux/Riemann integrable and
its integral is equal to 0.

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