Understanding Continuity at a Point

In calculus, the concept of continuity is fundamental. A function is considered continuous at a specific point if its graph can be drawn through that point without lifting your pen. This intuitive idea is formalized by a precise mathematical definition.

The Formal Definition of Continuity

For a function $f(x)$ to be continuous at a point $x = c$ , three conditions must be met:

A function is continuous at a point if it's defined, the limit exists, and the function value equals the limit.

Three key conditions must be satisfied for a function to be continuous at a specific point.

$f(c)$ must be defined. This means that the point $c$ must be in the domain of the function.
The limit of $f(x)$ as $x$ approaches $c$ must exist. This implies that the left-hand limit and the right-hand limit at $c$ must be equal: $\lim_{x\to c^-} f(x) = \lim_{x\to c^+} f(x)$ .
The limit of $f(x)$ as $x$ approaches $c$ must equal the function's value at $c$ . That is, $\lim_{x\to c} f(x) = f(c)$ .

If any one of these three conditions fails, the function is said to be discontinuous at $x = c$ .

Visualizing Continuity

Continuity at a point can be visualized by examining the behavior of a function's graph around that point. If there's a hole, a jump, or an asymptote at $x=c$ , the function is discontinuous there.

Consider a function $f(x)$ . For it to be continuous at $x=c$ , the graph must not have any breaks, jumps, or holes at that specific x-value. This means that as you approach $c$ from the left and from the right, the function's value should smoothly lead into the actual value of the function at $c$ . Think of it like a perfectly smooth road at a specific point – no bumps, no gaps, and the road surface at that point is exactly where you expect it to be based on the surrounding road.

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Types of Discontinuities

When a function is not continuous at a point, it exhibits a discontinuity. The most common types encountered in competitive exams are:

Type of Discontinuity	Condition Violated	Description
Removable Discontinuity	Limit exists, but $f(c)$ is undefined or $\lim_{x\to c} f(x) \neq f(c)$	A 'hole' in the graph that can be 'filled' by redefining the function at that point.
Jump Discontinuity	Left-hand limit $\neq$ Right-hand limit	The graph 'jumps' from one value to another at $x=c$ .
Infinite Discontinuity	Limit does not exist (approaches $\pm\infty$ )	Occurs at vertical asymptotes.

What are the three conditions for a function

f(x)

to be continuous at a point

x=c

$f(c)$ is defined. 2. $\lim_{x\to c} f(x)$ exists. 3. $\lim_{x\to c} f(x) = f(c)$ .

Applying the Definition in Problems

When solving problems, systematically check each of the three conditions. For piecewise functions, pay close attention to the points where the function definition changes, as these are common locations for discontinuities.

\lim_{x\to 2} f(x) = 5

and

f(2) = 5

, what can you conclude about the continuity of

f(x)

x=2

If $f(2)$ is defined and equals 5, and the limit as $x$ approaches 2 is also 5, then the function is continuous at $x=2$ (assuming the limit exists).

Learning Resources

Continuity of Functions - Khan Academy(video)

This video provides a clear explanation of the definition of continuity and demonstrates how to check for continuity at a point.

Continuity at a Point - Mathematics LibreTexts(documentation)

A comprehensive text resource detailing the definition of continuity, types of discontinuities, and theorems related to continuity.

Understanding Continuity - Brilliant.org(blog)

Explains the concept of continuity with interactive examples and intuitive explanations, focusing on the conditions for continuity.

Calculus: Continuity - Paul's Online Math Notes(documentation)

Detailed notes covering the definition of continuity, continuity at a point, and examples of continuous and discontinuous functions.

Continuity - Wolfram MathWorld(documentation)

A technical definition and properties of continuity, including formal mathematical statements and related concepts.

Continuity and Discontinuity - YouTube (Professor Leonard)(video)

A thorough video lecture explaining continuity at a point, including common pitfalls and examples relevant to calculus exams.

JEE Main Mathematics: Limits and Continuity - Byju's(blog)

This resource focuses on the JEE syllabus for limits and continuity, providing definitions and problem-solving strategies.

Continuity of Functions - University of Waterloo CEMC(documentation)

A PDF document with clear explanations and examples of continuity at a point, suitable for exam preparation.

What is a Continuous Function? - YouTube (3Blue1Brown)(video)

An intuitive and visually driven explanation of continuity, focusing on the geometric interpretation and the underlying ideas.

Continuity at a Point - Practice Problems - Brilliant.org(blog)

Offers practice problems with solutions to test your understanding of the definition of continuity at a point.

Definition of Continuity at a Point