Page 14 - DMTH505_MEASURE_THEOREY_AND_FUNCTIONAL_ANALYSIS

Unit 1: Differentiation and Integration: Differentiation of Monotone Functions

Notes
Example: Let f be a function defined by f (0) = 0 and f (x) = x sin (1/x) for x 0. Find
+
D f (0), D f (0), D f (0), D f (0).
–
+ –
1
hsin 0
f(0 h) f(o) h
D f (0) = Lim Lim
+
h o h h o h
1 1
= Lim sin 1, as 1 sin 1
h o h h
f(0 h) f(0) 1
Also D f (0) = Lim Lim sin 1
+ h o h h o h
1
( h)sin 0
f(0 h) f(0) h
–
D f (0) = Lim Lim
h o 0 h h o h
1
= Lim sin 1
h o h

f(0 h) f(0) 1
and D f (0) = Lim Lim sin 1
–
h o h h o h
Theorem: Let x be a Lebesgue point of a function f (t); then the indefinite integral

x
F (x) = F (a) + f(t) dt
a
is differentiable at each point x and F (x) = f (x).
Proof: Given that x is a Lebesgue point of f (t), so that

1 x h
Lim f(t) f(x) dt = 0 … (i)
h o h x
1 x h 1 x h 1
Now f(x) dt = f(x) 1 dt f(x)[t] x h
x
h x h x h
1
= f(x).h f(x)
h
1 x h
Thus f (x) = f(x) dt … (ii)
h x
x h x
Also F (x + h) – F (x) = f(t) dt f(t) dt
a a
x x h x x h
= f(t) dt f(t) dt f(t)dt f(t) dt
a x a x

F(x h) F(x) 1 x h
= f(t)dt … (iii)
h h x

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