Continuity and Average Rate of Change in Business Calculus

Continuity of Functions

Definition of Continuity at a Point

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

1. limx→c f(x) exists

2. f(c) exists

3. limx→c f(x) = f(c)

If any of these conditions fail, the function is not continuous at that point.

Example: Testing Continuity at a Point

Consider f(x) = √x. Is f(x) continuous at x = -2?

• Step 1: Check if f(-2) exists. f(-2) = √(-2) is not a real number, so f(-2) does not exist.

• Conclusion: The function fails condition #2. Therefore, f(x) is not continuous at x = -2.

Continuity on Intervals

• Open Interval (a, b): A function f(x) is continuous on an open interval (a, b) if it is continuous at every x-value in that interval.

• Closed Interval [a, b]: A function f(x) is continuous on a closed interval [a, b] if:

  • It is continuous on the open interval (a, b)

  • limx→a+ f(x) = f(a)

  • limx→b- f(x) = f(b)

Visuals: Open intervals are denoted with parentheses ( ), and closed intervals with brackets [ ].

Piecewise Functions and Continuity

Finding Parameters for Continuity

To ensure a piecewise function is continuous at a point where the formula changes, set the left and right limits equal at that point.

Example: Find the value of k to make f(x) continuous for all x

Given:

• f(x) = x2 - 3 for x < 2

• f(x) = kx + k for x ≥ 2

Set the pieces equal at x = 2:

• Left: f(2) = 22 - 3 = 4 - 3 = 1

• Right: f(2) = k(2) + k = 2k + k = 3k

• Set equal: 1 = 3k ⇒ k = 1/3

Average Rate of Change

Definition

The average rate of change of a function f(x) from x = a to x = b is given by:

$\text{Average Rate of Change} = \frac{f(b) - f(a)}{b - a}$

This measures the change in the function's value per unit change in x, and is interpreted as the slope of the secant line connecting the points (a, f(a)) and (b, f(b)).

Examples

• Example 1: For f(x) = x2, find the average rate of change from x = 1 to x = 2:

$\frac{f(2) - f(1)}{2 - 1} = \frac{2^2 - 1^2}{2 - 1} = \frac{4 - 1}{1} = 3$

• Example 2: For f(x) = x^2, find the average rate of change from x = 1 to x = 1 + h:

$\frac{f(1 + h) - f(1)}{(1 + h) - 1} = \frac{(1 + h)^2 - 1^2}{h} = \frac{1 + 2h + h^2 - 1}{h} = \frac{2h + h^2}{h} = 2 + h$

Warning: Always use the correct difference in the denominator (here, h, not 1 + h).

Applications

• Average rate of change is used in business calculus to measure things like average cost, average revenue, or average profit over an interval.

• It is also foundational for understanding the concept of the derivative, which measures instantaneous rate of change.