Functions of Bounded Variations Real Analysis part 7

Functions of bounded variations , Absolute continuity and functions of bounded variation, Real Analysis , Vitali Cover

What you’ll learn
✓ Functions of bounded variations Detailed Concepts
✓ Vitali’s Cover and its lemma , positive , negative and total variation
✓ When the function is Indefinite Integral and its absolute continuity
✓ Function is of bounded variation and its concept of Absolute Continuity

Requirements
● Basic concepts of Real Analysis , boundedness of functions and indefinite integrals, Fundamental theorem of Calculus

Description
In this 4 hour 10 min Course on Functions of Bounded Variation of Real Analysis part 7 , the contents included are

Differentiation of monotone functions

Definition of Vitali’s Cover, Lemma on Vitali covering theorem

The definition of derivatives of function f at x

If f is an increasing real valued function on the interval[a,b] then f is differentiable almost everywhere.

Functions of bounded variation

Positive variation, Negative variation, total variation

Jordan’s theorem: A function f is of bounded variation on closed interval a and b if and only if f is the difference of two monotone real valued functions on closed interval a and b.

If f is of bounded variation on closed interval a and b then derivative of f at x exists for almost all x in closed interval a and b.

The conditions for which f is a continuous function on closed interval a and b then there exist c belonging to that interval,

Differentiation of an integral

In this section we shall show that the derivative of indefinite integral of an integrable function is equal to the integrand almost everywhere. We begin by establishing some lemmas.

Very Important : Lemma on integrable function on closed interval then function if defined is a continuous function of bounded variation on closed interval.

The lemma on integrable function on closed interval and inetgration of f is 0 with limits x and t for all x belonging to closed interval then f(t) = 0 almost everywhere in closed interval a and b.

The Lemma on bounded and measurable functions.

The Theorems on integrable functions.

Absolute Continuity

Definition of Absolutely continous.

Lemma on Absolutely continuous on closed interval.

Corollary on absolutely continuous.

Lemma on Absolute continuous on closed interval and derivative of f is 0.

Definition of indefinite integral.

Theorem on indefinite integral.

Corollary on absolute continuous function.

Including detailed concept and theorems used.

Who this course is for
■ Msc.maths, Bsc. maths, Engineering students, graduate and post graduate students, ugc net exam and other comeptitive exams

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