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Theory of Real Functions Notes

Topics in our Theory of Real Functions Notes PDF

In these “Theory of Real Functions Notes PDF”, you will study the study of real valued functions that would develop an analytical ability to have a more matured perspective of the key concepts of calculus, namely, limits, continuity, differentiability and their applications.

The topics we will cover will be taken from the following list:

Limits of Functions: Limits of functions (ε δ − approach), Sequential criterion for limits, Divergence criteria, Limit theorems, One-sided limits, Infinite limits and limits at infinity.

Continuous Functions and their Properties: Continuous functions, Sequential criterion for continuity and discontinuity, Algebra of continuous functions, Properties of continuous functions on closed and bounded intervals; Uniform continuity, Non-uniform continuity criteria, Uniform continuity theorem.

Derivability and its Applications: Differentiability of a function, Algebra of differentiable functions, Carathéodory’s theorem, Chain rule; Relative extrema, Interior extremum theorem, Rolle’s theorem, Mean- value theorem and applications, Intermediate value property of derivatives, Darboux’s theorem.

Taylor’s Theorem and its Applications: Taylor polynomial, Taylor’s theorem with Lagrange form of remainder, Application of Taylor’s theorem in error estimation; Relative extrema, and to establish a criterion for convexity; Taylor’s series expansions of ex , sin x and cos x.


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