**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 e^{x} , sin x and cos x.

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