# Theory of Real Function

By Dr. Anjaly Kishore   |   Vimala College, Thrissur, Kerala
Learners enrolled: 549
The course “Theory of Real Functions” is proposed for B.Sc. (Hons) Mathematics as the Core course. The course is an indispensable part of pure and applied mathematics which equip the undergraduate students to have a deeper and thorough understanding of the Functions of Real Variables and to apply the concepts of Limits, Continuity and Differentiability in real world problems. The course includes the introduction to Real Functions, Limits, Continuity, Uniform Continuity,  Differentiability, Intermediate Value Property, Mean Value Theorems, Taylor’s and Maclaurins’ Series expansions, Riemann integration and Improper integrals. The primary Learning Outcome of this course is Quantitative Reasoning, which is to understand and apply mathematical concepts, analyze and interpret various types and properties of Real Functions. The Learning Outcomes will be assessed through targeted questions - either the comprehensive final or an outside assignment. The applications of mathematics in various disciplines are seeking more analytical skills and deeper understanding of Real Functions. This course enable the  students who are interested in Economics, Physics, Engineering and Business Management to develop the skills in  analysing and solving  different types of functions of real variables which emerge in  mathematical modelling of real world problems.
Summary
 Course Status : Ongoing Course Type : Core Duration : 16 weeks Category : Mathematics Credit Points : 5 Level : Undergraduate Start Date : 08 Jul 2024 End Date : 12 Oct 2024 Enrollment Ends : 31 Aug 2024 Exam Date : 15 Dec 2024 IST Exam Shift : Shift 1

Note: This exam date is subject to change based on seat availability. You can check final exam date on your hall ticket.

### Course layout

Weeks  Weekly Lecture Topics (Module Titles) Assignment

Day 1 Module 1 : Introduction to Real Functions No. 1 MCQ
Day 2 Module 2 : Limits: Definition and Properties
Day 3 Module 3 : Sequential Criterion for Limits
Day 4 Interaction based on the three modules covered
Day 5 Assignment

Day 1 Module 4 : Limit Theorems: Part I
Day 2 Module 5 : Limit Theorems: Part II
Day 3 Module 6 : One sided Limits
Day 4 Interaction based on the three modules covered
Day 5 Assignment

Day 1 Module 7 : Infinite limits
Day 2 Module 8 : Limits at infinity
Day 3 Module 9 : Continuous Functions
Day 4 Interaction based on the three modules covered
Day 5 Assignment

Day 1 Module 10 : Combinations of Continuous Functions
Day 2 Module 11 : Composition of Continuous Functions
Day 3 Module 12 : Continuous Functions on Intervals, Boundedness theorem
Day 4 Interaction based on the three modules covered
Day 5 Assignment

5
Day 1 Module 13 : Maximum Minimum Theorem
Day 2 Module 14 : Location of Roots Theorem
Day 3 Module 15 : Bolzano’s Intermediate Value Theorem
Day 4
Day 5 Assignment

Day 1 Module 16 : Uniform Continuity
Day 2 Module 17 : Lipschitz Function
Day 3 Module 18 : Continuous Extension Theorem
Day 4 Interaction based on the three modules covered
Day 5 Assignment

Day 1 Module 19 : Approximations No.
Day 2 Module 20 : Monotone Functions
Day 3 Module 21 : Inverse Functions
Day 4 Interaction based on the three modules covered
Day 5 Assignment

Day 1 Module 22 : Differentiability: Introduction to Derivatives
Day 2 Module 23: Tangent lines and Rates of Change
Day 3 Module 24 : Chain Rule and Caratheodery’s theorem
Day 4 Interaction based on the three modules covered
Day 5 Assignment

9
Day 1 Module 25 : Derivatives of Inverse Functions
Day 2 Module 26 : Role of Derivatives in Real World
Day 3 Module 27: Differentials and Linear approximations
Day 4 Interaction based on the three modules covered
Day 5 Assignment

10
Day 1 Module 28 : Extrema of Functions
Day 2 Module 29 : Rolle’s theorem &amp; Mean Value Theorem
Day 3 Module 30 : Applications of Mean Value Theorem
Day 4
Day 5 Assignment

11
Day 1 Module 31 : Intermediate Value Property, Darboux’s Theorem
Day 2 Module 32 : Increasing and Decreasing Functions
Day 3 Module 33: L’ Hôpital’s Rule
Day 4 Interaction based on the three modules covered
Day 5 Assignment

12
Day 1 Module 34 : Concavity and Inflection Points
Day 2 Module 35 : Cauchy’s Mean Value Theorem
Day 3 Module 36 : Taylor’s theorem
Day 4 Interaction based on the three modules covered
Day 5 Assignment

13
Day 1 Module 37 : Applications of Taylor’s theorem, Newton’s Method No. 13 MCQ
Day 2 Module 38 : Taylor’s Series
Day 3 Module 39 : Maclaurin’s Series
Day 4 Interaction based on the three modules covered
Day 5 Assignment

14
Day 1 Module 40 : Applications of Taylor’s and Maclaurins’ series
Day 2 Module 41 : Partitions and Riemann Sums
Day 3 Module 42 : Properties of Riemann integrals
Day 4
Day 5 Assignment

15
Day 1 Module 43 : Existence of Riemann Integrals
Day 2 Module 44 : Riemann Integrable Functions
Day 3 Module 45 : Additivity Theorem
Day 4 Interaction based on the three modules covered
Day 5 Assignment

16
Day 1 Module 46 : Fundamental Theorem of Calculus
Day 2 Module 47 : Substitution Theorem
Day 3 Module 48 : Improper Integrals
Day 4 Interaction based on the three modules covered
Day 5 Assignment

17
Day 1 Module 49 : Applications of Improper Integrals
Day 2 Module 50 : Cauchy Principal Value and Tests for convergence
Day 3 Revision
Day 4
Day 5 Assignment

### Books and references

Robert G Bartle & Donald R Sherbert - Introduction to Real Analysis 4th  edition- Wiley
Terrence Tao - Analysis I, 3rd edition- Texts and Readings in Mathematics

### Dr. Anjaly Kishore

Vimala College, Thrissur, Kerala
Dr Anjaly Kishore (Course Coordinator) has more than 10 years of teaching experience in handling the topics Real Analysis, Calculus, Number Theory, Logic and Algebra. She has more than 15 publications in reputed peer reviewed journals.

Research Guide under the University of Calicut since 2018.
Member, Board of Studies, Vimala College.
Joint Coordinator, IQAC and NAAC Core committee, Vimala College.
Former Joint Coordinator, Career Guidance & Placement Cell, Vimala College
Reviewer of Various Journals
Resource person at Various Colleges.
Delivered lectures for DTH programmes.