Answer to Question #60679 in Real Analysis for esita

Question

a) Consider the function f (x) = 2cos x in the interval 





2
,0
π
. Show that
( , ) ( , )
1 2 L P f ≤ L P f and ) ( , ) ( ,
2 1 U P f ≤ U P f where 





=
2
,
3
1
,0
π π
P and






=
2
,
3
,
6
2
,0
π π π
P . (6)
b) Show that the derivative f ′ of the following function f given by





=
≠
=
0 if 0
if 0
1
sin ( )
2
x
x
x
x
f x
exists at x = 0 but f ′ is not continuous at 0 .