M2: Analysis II - Continuity and Differentiability (2024-25)
Main content blocks
- Lecturer: Profile: Paul Balister
Course information
Course term: Hilary
Course lecture information: 16 lectures
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.
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. Definition of continuity of functions on subsets of \(\mathbb{R}\) and \(\mathbb{C}\) in terms of \(\varepsilon\) and \(\delta\). 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 Continuous 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'Hôpital's Formula. Taylor's Theorem with remainder in Lagrange's form; examples. The binomial expansion with arbitrary index.
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'Hôpital's Formula. Taylor's Theorem with remainder in Lagrange's form; examples. The binomial expansion with arbitrary index.
Section outline
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Supplementary notes on exp(x) (non-examinable). Uses material from Analysis I only.
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Construction of a space filling curve (non-examinable). Uses material from the course on uniform convergence.
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Elementary (non-examinable) proof of ∑1/n2 = π2/6. Later parts use Mean Value Theorem.
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