Page 12 - DMTH505_MEASURE_THEOREY_AND_FUNCTIONAL_ANALYSIS

Unit 1: Differentiation and Integration: Differentiation of Monotone Functions

Notes

Notes
+
1. D f (x) D f (x) and Df(x) Df(x)
+
+
If D f (x) = D f (x), then we conclude that right hand derivative of f (x) exists at the
+
+
–
point x and denoted by f (x ). Similarly if D f (x) = D f (x), we say that f (x) is left
–
–
differentiable at x and denote this common value by f (x ).
2. The function is said to be differentiable at x if all the four Dini’s derivatives are equal
but different than , i.e. if
+
–
D f (x) = D f (x) = D f (x) = D f (x)
+ –
and their common value is denoted by f (x).
Properties of Dini's Derivatives
1. Dini’s derivatives always exist, may be finite or infinite for every function f.
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+
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2. D (f + g) D f + D g with similar properties for the other derivatives.
3. If f and g are continuous at a point ‘x’, then
+
+
+
D (f . g) (x) f (x) D g (x) + g (x) D f (x).
+
4. D f (x) = – D (– f (x))
+
–
and D f (x) = – D (– f (x)).
–
+
5. If f is a continuous function on [a, b] and one of its derivatives (say D ) is non-negative on
(a, b). Then f is non-decreasing on [a, b] i.e.
f (x) f (y) whenever x y, y [a, b].
6. If f is any function on an interval [a, b], then the four derivatives if exist are measurable.
1.1.5 Lebesgue Differentiation Theorem

Statement: Let f : [a, b] R be a finite valued monotonically increasing function, then f is
differentiable. Also f : [a, b] R is L-integrable and
b
f (x) dx f(b) f(a) .
a
Proof: Define a sequence <f > of non-negative functions, where f : [a, b] R such that,
n n
1
f (x) = n f x f(x) , x [a, b] … (1)
n n
and set f (x) = f (b), for x b.
By hypothesis, f : [a, b] R is an increasing function, therefore f : [a, b] R is also an increasing
n
function and hence integrable in the Lebesgue sense.
Again from (i) we have
f{x (1/n)} f(x)
Lim f (x) = Lim , x [a, b] ,
n n 1/n 0 (1/n)

= f (x), a.e.

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