# M2: Analysis II - Continuity and Differentiability - Material for the year 2019-2020

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16 lectures

In this term's lectures, we study continuity of functions of a real or complex variable, and differentiability of functions of a real variable.

At the end of the course students will be able to apply limiting properties to describe and prove continuity and differentiability conditions for real and complex functions. They will be able to prove important theorems, such as the Intermediate Value Theorem, Rolle's Theorem and Mean Value Theorem, and will continue the study of power series and their convergence.

See the examinable syllabus.

Definition of the function limit. Definition of continuity of functions on subsets of $\mathbb{R}$ and $\mathbb{C}$ in terms of $\varepsilon$ and $\delta$. Continuity of real valued functions of several variables. The algebra of continuous functions; examples, including polynomials. Intermediate Value Theorem for continuous functions on intervals. Boundedness, maxima, minima and uniform continuity for continuous functions on closed intervals. Monotone functions on intervals and the Inverse Function Theorem.

Sequences and series of functions, uniform convergence. Weierstrass's M-test for uniformly convergent series of functions. Uniform limit of a sequence of continuous functions is continuous. Continuity of functions defined by power series.

Definition of the derivative of a function of a real variable. Algebra of derivatives, examples to include polynomials and inverse functions. The derivative of a function defined by a power series is given by the derived series (proof not examinable). Vanishing of the derivative at a local maximum or minimum. Rolle's Theorem, Mean Value Theorem, and Cauchy's (Generalized) Mean Value Theorem with applications: Constancy Theorem, monotone functions, exponential function and trigonometric functions. L'HĂ´pital's Formula. Taylor's Theorem with remainder in Lagrange's form; examples. The binomial expansion with arbitrary index.

1) Lecture Notes for this course.

2) W. Rudin, *Principles of Mathematical Analysis* (McGraw-Hill, Third Edition), Chapters 4, 5, 7.

3) T. M. Apostol, *Mathematical Analysis* (Addison-Wesley Pub. Company), Chapters 4 and 5.

4) M. Spivak, *Calculus* (Cambridge University Press; 3 edition), Sections 5 to 12.

1) M. Giaquinta and G. Modica, *Mathematical Analysis: Functions of One Variable: v. 1* (BirkhĂ¤user).

2) V. A. Zorich, *Mathematical Analysis I* (2nd Edition, Universitext, Springer).