Course Overview: In this term's lectures, we study continuity of functions of a real or complex variable, and differentiability of functions of a real variable.
Course Syllabus:
See the examinable syllabus.
Lecturer(s):
Prof. Zhongmin Qian
Learning Outcomes: 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.
Course Synopsis: Definition of the function limit. Examples and counter examples to illustrate when \(\lim_{x\rightarrow a}f\left( x\right) =f\left( a\right) \) (and when it doesn't). 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. Continuous functions on closed bounded intervals: boundedness, maxima and minima, uniform continuity. Intermediate Value Theorem. Inverse Function Theorem for continuous strictly monotone functions.
Sequences and series of functions. Uniform limit of a sequence of continuous functions is continuous. Weierstrass's M-test for uniformly convergent series of functions. 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 with simple applications: constant and monotone functions. Cauchy's (Generalized) Mean Value Theorem and L'Hôpital's Formula. Taylor's Theorem with remainder in Lagrange's form; examples. The binomial expansion with arbitrary index.
