Continuity of Functions: Definitions, Properties, and Applications

Continuity of Functions

Understanding Continuity at a Point

Continuity is a fundamental concept in calculus, describing the behavior of functions at specific points and over intervals. A function is said to be continuous at a point if its graph does not have any breaks, jumps, or holes at that point.

• Definition: A function f(x) is continuous at x = c if all three of the following conditions are satisfied:

  1. f(c) exists: The function is defined at x = c.

  2. \( \lim_{x \to c} f(x) \) exists: The limit as x approaches c exists.

  3. \( \lim_{x \to c} f(x) = f(c) \): The value of the function at c equals the limit as x approaches c.

• Graphical Interpretation: The graph of y = f(x) does not break at x = c; the point (c, f(c)) is approached and reached smoothly.

Examples of Continuity and Discontinuity

Consider the following functions and their behavior at x = 2:

• \( f(x) = \frac{x^2 - 4}{x - 2} \): The limit exists at x = 2, but f(2) is undefined (hole in the graph).

• \( g(x) = \begin{cases} x + 2 & x \neq 2 \\ 3 & x = 2 \end{cases} \): Both the limit and g(2) exist, but they are not equal (removable discontinuity).

• \( h(x) = \frac{|x - 2|}{x - 2} \): Neither the limit nor h(2) exists (jump discontinuity).

• \( k(x) = x + 2 \): Both the limit and k(2) exist and are equal; the function is continuous at x = 2.

Types of Discontinuities

• Removable Discontinuity: The limit exists, but the function value is either undefined or not equal to the limit (e.g., f(x) and g(x) above).

• Jump Discontinuity: The left and right limits at a point are not equal (e.g., h(x) and q(x) = \begin{cases} \frac{x-2}{|x-2|} & x \neq 2 \\ 1/2 & x = 2 \end{cases}).

• Infinite Discontinuity: The function approaches infinity at a point, often due to a vertical asymptote (e.g., r(x) = \frac{1}{x-2} at x = 2).

Piecewise Functions and Real-World Example

Piecewise functions can model real-world scenarios with discontinuities. For example, in 'just in time inventory control,' inventory is replenished at regular intervals, resulting in a graph with jump discontinuities at each restocking point.

• Example: Inventory function for monthly deliveries of 10 units:

  • For 0 ≤ x < 1: f(x) = -10x + 10

  • For 1 ≤ x < 2: f(x) = -10(x-1) + 10

  • For 2 ≤ x < 3: f(x) = -10(x-2) + 10

  • ...and so on.

Continuity of Polynomial and Common Functions

• Polynomials: Every polynomial function is continuous everywhere on the real line. That is, for any real number c, f(c) is defined and \( \lim_{x \to c} f(x) = f(c) \).

• Rational Functions: Continuous everywhere except where the denominator is zero.

Algebraic Properties of Continuous Functions

If f(x) and g(x) are continuous at x = c, then the following functions are also continuous at x = c:

Operation	Resulting Function
Sum	\( h(x) = f(x) + g(x) \)
Difference	\( h(x) = f(x) - g(x) \)
Constant Multiple	\( h(x) = k \cdot f(x) \), where k is a constant
Product	\( h(x) = f(x) \cdot g(x) \)
Quotient	\( h(x) = \frac{f(x)}{g(x)} \), provided g(c) \neq 0
Power	\( h(x) = [f(x)]^n \), for any whole number n \geq 1
Root	\( h(x) = \sqrt[n]{f(x)} \), provided \sqrt[n]{f(c)} is real

Continuity on Intervals

• Definition: A function f(x) is continuous on an interval (a, b) if it is continuous at every point c in (a, b).

• Graphical Meaning: The graph of y = f(x) has no breaks on (a, b).

• Endpoint Continuity: For closed intervals, continuity at endpoints is defined using one-sided limits:

  • Right continuity at a: \( \lim_{x \to a^+} f(x) = f(a) \)

  • Left continuity at b: \( \lim_{x \to b^-} f(x) = f(b) \)

Intermediate Value Theorem (IVT)

The Intermediate Value Theorem is a key result about continuous functions:

• If f(x) is continuous on an interval containing a and b (with a < b), and f(a) and f(b) have opposite signs, then there exists at least one value r in [a, b] such that f(r) = 0.

• This means the graph of f(x) crosses the x-axis somewhere between a and b.

Application: Approximating Roots Using IVT

The IVT can be used to approximate roots of equations, such as square roots.

• Example: To approximate \( \sqrt{2} \), consider f(x) = x^2 - 2.

  • For a = 1.4, f(1.4) = -0.04; for b = 1.5, f(1.5) = 0.25. Thus, 1.4 < \sqrt{2} < 1.5.

  • Refining further: f(1.41) = -0.0119, f(1.42) = 0.0164; so 1.41 < \sqrt{2} < 1.42.

  • Further refinement: f(1.414) = -0.0006, f(1.415) = 0.0022; so 1.414 < \sqrt{2} < 1.415.

• Result: \( \sqrt{2} \approx 1.4142135623730950488 \) (to 20 decimal places).

Additional info: The process described above is known as the bisection method, a numerical technique for finding roots of continuous functions by repeatedly narrowing the interval where the function changes sign.