Continuity of Functions - Single-Variable Calculus (Undergraduate Foundation)

This course provides a focused and rigorous examination of continuity, a fundamental property of functions that underpins the whole of differential and integral calculus. We move from an intuitive understanding of "unbroken" graphs to the formal, limit-based definition of continuity. The course systematically explores the different types of discontinuities and the powerful theorems that apply only to continuous functions. A command of continuity is essential for understanding why calculus works. This concept ensures that functions are predictable and well-behaved, a necessary condition for modeling real-world phenomena and for the validity of the major theorems of calculus. It is the bridge between the concept of a limit and the concept of a derivative, explaining why a function must be continuous to be differentiable. By the end of this course, you will be able to use the three-part definition to test for continuity at a point, identify and classify removable, jump, and infinite discontinuities, and apply the Intermediate Value Theorem and the Extreme Value Theorem to analyse function behavior on a closed interval. This course is designed for first-year undergraduates in science, technology, engineering, and mathematics who have completed a course on limits. It is a critical prerequisite for the study of differentiability and is essential for any student seeking a deep understanding of calculus theory.

5 hrs

Payment required for enrolment

Enrolment valid for 12 months

This course is also part of the following learning tracks. You may join a track to gain comprehensive knowledge across related courses.

[OAU, Ife] MTH 201: Mathematical Methods I

This learning track delivers the complete mathematical toolkit required for a university-level science, engineering, or computing degree. It systematically covers the entire MTH 201 curriculum, building from the foundational principles of single-variable calculus - functions, limits, continuity, and differentiability - to the advanced methods of multivariable calculus, infinite series, numerical methods, and ordinary differential equations. This is the definitive preparation for advanced quantitative study. This programme is designed for second-year students offering MTH 201 at Obafemi Awolowo University, Ile-Ife, Nigeria. It is also helpful for any student in a STEM field - including physics, engineering, and computer science - who requires a rigorous and comprehensive command of calculus and its applications. This track delivers a full skill set in mathematical analysis and applied problem-solving. Graduates will be able to solve a wide range of problems, from optimising multivariable functions to modelling dynamic systems with differential equations and testing the convergence of infinite series. This programme directly prepares students for success in advanced courses in vector calculus, partial differential equations, and real analysis, providing the necessary foundation for a career in engineering, data science, or theoretical physics.

[NUC Core] GET 209: Engineering Mathematics I

Master the mathematical language of engineering. This programme delivers the complete analytical toolkit required for a successful engineering career, covering single-variable calculus, multivariable calculus, linear algebra, and vector analysis. It provides the essential foundation for all subsequent engineering courses. This programme is for second-year undergraduate students across all engineering disciplines. It delivers the official NUC CCMAS curriculum for Engineering Mathematics, providing the core training required for advanced modules in mechanics, thermodynamics, and circuit theory. Model and analyse complex physical systems using calculus, linear algebra, and vector analysis. You will be equipped to solve problems in dynamics, statics, and field theory, providing the quantitative proficiency required for advanced engineering study and professional practice.

[UI, Ibadan] MAT 223: Real Analysis

This learning track provides the complete theoretical machinery of single-variable calculus and analysis. We build the subject from first principles, establishing the rigorous logical framework required for advanced quantitative disciplines. This is the 'why' behind the mathematics that powers science and engineering. This track is built for second-year engineering and physical science students, particularly those at the University Of Ibadan. It is also structured for any student requiring the same rigorous theoretical foundation for advanced quantitative study. On completion, you will command the complete theoretical basis of single-variable calculus. You will construct formal proofs, rigorously analyse function behaviour, and determine the convergence of infinite series. This programme provides the non-negotiable prerequisite knowledge for advanced study in differential equations, complex analysis, and theoretical physics.

Course Chapters

1. Introduction

Continuity (at an interior point, at an endpoint, on an interval) of real-valued functions - graphical illustration, formal and informal definitions.

Chapter lessons

1-1. Informal definition

16:07

Informal definition of continuity.

1-2. Continuity at an interior point

13:34

Continuity at an interior point of the domain of a function.

1-3. Continuity at an endpoint

8:53

Continuity at an endpoint of the domain of a function.

1-4. Continuity on an interval

17:22

Continuity of a function on an interval in its domain.

2. Continuous Functions

Examples of continuous functions, worked problems on continuity.

Chapter lessons

2-1. Examples of continuous functions

24:12

Some examples of functions that are continuous everywhere in their domain of definition.

3. Types of Discontinuity

Various types of discontinuities - meaning and examples.

Chapter lessons

3-1. Removable discontinuities

10:52

Meaning of removable and non-removable discontinuities.

4. Theorems on Continuous Functions

Understanding the max-min theorem and the intermediate-value theorem.

Chapter lessons

4-1. The max-min theorem

11:12

Understanding the max-min theorem.

4-2. The intermediate-value theorem

11:20

Understanding the intermediate-value theorem.