Defining Continuity at a Point

Introduction

Key Concepts

Definition of Continuity at a Point

In calculus, a function \( f(x) \) is said to be continuous at a point \( x = c \) if three specific conditions are satisfied. These conditions ensure there are no abrupt jumps, breaks, or holes in the function at that point. Formally, the function \( f(x) \) is continuous at \( c \) if:

1. Existence of \( f(c) \): The function must be defined at \( x = c \). In other words, \( f(c) \) exists.
2. Existence of \( \lim\limits_{x \to c} f(x) \): The limit of \( f(x) \) as \( x \) approaches \( c \) must exist.
3. Equality of Limit and Function Value: The limit of \( f(x) \) as \( x \) approaches \( c \) must equal the function value at that point, i.e., \( \lim\limits_{x \to c} f(x) = f(c) \).

Mathematically, this can be expressed as: $$ \text{Continuous at } x = c \quad \text{if} \quad \lim\limits_{x \to c} f(x) = f(c) $$

Understanding Limits in Continuity

The concept of limits is pivotal in defining continuity. The limit \( \lim\limits_{x \to c} f(x) \) signifies the value that \( f(x) \) approaches as \( x \) approaches \( c \). For continuity, this approaching value must align perfectly with the actual function value at \( c \).

Consider the function \( f(x) = \frac{x^2 - 4}{x - 2} \). At \( x = 2 \), the function appears undefined due to division by zero. However, simplifying \( f(x) \) gives: $$ f(x) = \frac{(x - 2)(x + 2)}{x - 2} = x + 2 \quad \text{for} \quad x \neq 2 $$ Here, the limit as \( x \) approaches 2 is: $$ \lim\limits_{x \to 2} f(x) = 4 $$ But since \( f(2) \) is undefined, \( f(x) \) is not continuous at \( x = 2 \).

Types of Discontinuities

Understanding the types of discontinuities helps in analyzing functions comprehensively. The primary types are:

• Removable Discontinuity: Occurs when \( \lim\limits_{x \to c} f(x) \) exists, but \( f(c) \) is either not defined or not equal to the limit. This type can be "removed" by redefining \( f(c) \) to equal the limit.
• Jump Discontinuity: Happens when the left-hand limit and the right-hand limit exist but are not equal. The function "jumps" from one value to another at \( c \).
• Infinite Discontinuity: Arises when the limit \( \lim\limits_{x \to c} f(x) \) does not exist because it approaches infinity or negative infinity.

Continuity and Differentiability

While continuity is a prerequisite for differentiability, the converse is not always true. If a function is differentiable at \( c \), it must be continuous there. However, a function can be continuous at a point but not differentiable. A classic example is \( f(x) = |x| \) at \( x = 0 \), where the function is continuous but has a sharp corner, making it non-differentiable at that point.

Examples of Continuous and Discontinuous Functions

Continuous Function Example:

Consider \( f(x) = 3x + 2 \). This linear function is continuous for all real numbers. For any \( c \), \( \lim\limits_{x \to c} f(x) = 3c + 2 = f(c) \).

Discontinuous Function Example:

Take \( f(x) = \frac{1}{x} \). This function is discontinuous at \( x = 0 \) because \( f(0) \) is undefined, and the limits from the left and right approach infinity and negative infinity, respectively.

Graphical Interpretation of Continuity

Graphically, a function is continuous at \( x = c \) if you can draw the function at that point without lifting your pencil from the paper. Removable discontinuities appear as holes, jump discontinuities as sudden jumps, and infinite discontinuities as vertical asymptotes.

For instance, the graph of \( f(x) = x^2 \) is a smooth parabola, indicating continuity everywhere. In contrast, \( f(x) = \frac{1}{x} \) has a vertical asymptote at \( x = 0 \), illustrating an infinite discontinuity.

Continuous Functions on Intervals

A function can be continuous over an interval if it is continuous at every point within that interval. For example, \( f(x) = \sin(x) \) is continuous for all real numbers, making it continuous on any interval. On the other hand, a piecewise function may only be continuous on specific intervals depending on its definitions.

Properties of Continuous Functions

• Intermediate Value Theorem: If \( f \) is continuous on \([a, b]\) and \( N \) is between \( f(a) \) and \( f(b) \), then there exists at least one \( c \) in \((a, b)\) such that \( f(c) = N \).
• Extreme Value Theorem: If \( f \) is continuous on a closed interval \([a, b]\), then \( f \) attains a maximum and minimum value on that interval.

Limit Laws and Continuity

Several limit laws are applicable when dealing with continuous functions. If \( f \) and \( g \) are continuous at \( c \), then:

• The sum \( f + g \) is continuous at \( c \).
• The product \( f \cdot g \) is continuous at \( c \).
• The quotient \( \frac{f}{g} \) is continuous at \( c \) provided \( g(c) \neq 0 \).
• The composition \( f(g(x)) \) is continuous at \( c \) if \( g \) is continuous at \( c \) and \( f \) is continuous at \( g(c) \).

Continuity in Polynomial and Rational Functions

Polynomial functions, such as \( f(x) = x^3 - 2x + 1 \), are continuous everywhere on the real number line. Rational functions, which are ratios of polynomials like \( f(x) = \frac{x^2 - 1}{x - 1} \), are continuous wherever the denominator is not zero. Points where the denominator is zero indicate potential discontinuities.

Piecewise Functions and Continuity

Piecewise functions are defined by different expressions over different intervals. Ensuring continuity in piecewise functions involves checking the boundaries where the expressions meet. For example:

$$ f(x) = \begin{cases} x + 1 & \text{if } x < 2 \\ 3x - 5 & \text{if } x \geq 2 \end{cases} $$

To verify continuity at \( x = 2 \), compute the left-hand limit: $$ \lim\limits_{x \to 2^-} f(x) = 2 + 1 = 3 $$ And the right-hand limit: $$ \lim\limits_{x \to 2^+} f(x) = 3(2) - 5 = 1 $$ Since the limits are not equal, \( f(x) \) is discontinuous at \( x = 2 \).

Comparison Table

Summary and Key Takeaways