2-3 Linearity and Continuity of Graphs

Property

• Continuous Function: A function whose graph is an unbroken line or curve with no gaps, representing a continuous domain where inputs can be any real number within an interval.
• Discrete Function: A function whose graph consists of isolated, unconnected points, representing a discrete domain where inputs are specific, separate values (like integers).

Examples

• Continuous: The distance traveled by a car $d(t)$ over $t$ hours. The graph is an unbroken line because time and distance can take on any fractional value.
• Discrete: The total cost $C(n)$ of buying $n$ concert tickets. The graph consists of individual dots because you can only purchase whole numbers of tickets ($n = 1, 2, 3, \dots$).

Explanation

A continuous function represents data that can take on any value within an interval, resulting in a graph that can be drawn without lifting your pencil. In contrast, a discrete function represents data that can only take on specific, separate values, resulting in a graph of unconnected dots. Understanding the real-world context of the variables allows you to determine whether the function modeling the situation should be continuous or discrete.

Property

A continuous domain includes all real number values within an interval, meaning every point between two values is included. A discrete domain includes only specific, separate values — often integers or a limited set of numbers.

• Continuous domain example: all real numbers from $1$ to $10$, written as $1 \leq x \leq 10$
• Discrete domain example: only the whole number values $\{1, 2, 3, 4, 5\}$

Property

A graph (or function) is classified as neither continuous nor discrete when it contains both connected intervals (continuous segments) and isolated points that do not connect to those segments.

Property

When comparing functions directly, linear functions maintain constant rates of change while nonlinear functions have variable rates of change.
For any linear function $y = mx + b$, the rate of change between any two points is always $m$.
For nonlinear functions, the rate of change $\frac{\Delta y}{\Delta x}$ varies between different intervals.

Examples