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Theory of Real Functions

By Dr. Anjaly Kishore   |   St. Mary’s College (Autonomous), Thrissur
Learners enrolled: 62
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 : Upcoming
Course Type : Core
Language for course content : English
Duration : 12 weeks
Category :
  • Mathematics
Credit Points : 5
Level : Undergraduate
Start Date : 14 Jul 2025
End Date : 31 Oct 2025
Enrollment Ends : 31 Aug 2025
Exam Date :
NCrF Level   : 6.0

Page Visits



Course layout

Weeks  Weekly Lecture Topics  Assignment No.& Type

Day 1 Module 1 : Introduction to Real Functions 
Day 2 Module 2 : Limits: Definition and Properties
Day 3 Module 3 : Sequential Criterion for Limits
Day 4 Module 4 : Limit Theorems: Part I
Day 5 Assignment No. 1 MCQ

Day 1 Module 5 : Limit Theorems: Part II
Day 2 Module 6 : One sided Limits
Day 3 Module 7 : Infinite limits
Day 4 Module 8 : Limits at infinity
Day 5 Assignment No. 2 MCQ

Day 1 Module 9 : Continuous Functions
Day 2 Module 10 : Combinations of Continuous Functions
Day 3 Module 11 : Composition of Continuous Functions
Day 4 Module 12 : Continuous Functions on Intervals, 
         Boundedness theorem
Day 5 Assignment   No. 3 MCQ
 
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 Interaction based on the modules covered
Day 5 Assignment No. 4 MCQ

Day 1 Module 16 : Uniform Continuity
Day 2 Module 17 : Lipschitz Function
Day 3 Module 18 : Continuous Extension Theorem
Day 4 Module 19 : Approximations
Day 5 Assignment No. 5 MCQ

Day 1 Module 20 : Monotone Functions
Day 2 Module 21 : Inverse Functions
Day 3 Module 22 : Differentiability: Introduction to Derivatives
Day 4 Module 23: Tangent lines and Rates of Change
Day 5 Assignment No. 6 MCQ

Day 1 Module 24 : Chain Rule and Caratheodery’s theorem 

Day 2 Module 25 : Derivatives of Inverse Functions
Day 3 Module 26 : Role of Derivatives in Real World
Day 4 Module 27: Differentials and Linear approximations
Day 5 Assignment No. 7 MCQ

Day 1 Module 28 : Extrema of Functions
Day 2 Module 29 : Rolle’s theorem & Mean Value Theorem
Day 3 Module 30 : Applications of Mean Value Theorem
Day 4 Interaction based on the modules covered
Day 5 Assignment No. 8 MCQ

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 Module 34 : Concavity and Inflection Points
Day 5 Assignment No. 9 MCQ

10 Day 1 Module 35 : Cauchy’s Mean Value Theorem
Day 2 Module 36 : Taylor’s theorem
Day 3 Module 37 : Applications of Taylor’s theorem, Newton’s Method
Day 4 Module 38 : Taylor’s Series
Day 5 Assignment No. 10 MCQ

11 Day 1 Module 39 : Maclaurin’s Series
Day 2 Module 40 : Applications of Taylor’s and Maclaurins’ series
Day 3 Module 41 : Partitions and Riemann Sums
Day 4 Module 42 : Properties of Riemann integrals
Day 5 Assignment No. 11 MCQ

12 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 modules covered
Day 5 Assignment No. 12 MCQ

13 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 modules covered
Day 5 Assignment No. 13 MCQ

14 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 No. 14 MCQ

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

Instructor bio

Dr Anjaly Kishore (Course Coordinator) has more than 14
   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.
 Dean of Research, St. Mary’s College
 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.
 Member : FLAIR Kerala, ADMA

Course certificate

* Internal Assessment- Weekly assessments released in the course shall be considered for Internal Marks and will carry 30 percent for the Overall Result. Out of all weekly   assignments, the best/top five scores will be considered for the final Internal Assessment marks.

* End-term Assessment-The final exam shall be conducted by NTA, and will carry 70 percent for the overall Result.

* All students who obtain 40% marks in the internal assessment and 40% marks in the end-term proctored exam separately will be eligible for the SWAYAM Credit Certificate.


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