Continuity

Continuity is a fundamental concept in calculus that describes how “smooth” a function is. Put simply, a function is continuous if you can draw its graph without lifting your pencil off the paper. There are no jumps, holes, or breaks in the function at that point.

Why Care About Continuity?

Continuity is crucial because:

• It’s a requirement for a function to be differentiable (have a derivative) at a point
• Many real-world phenomena are modeled by continuous functions (like smooth motion, temperature changes)
• It’s essential for integration and solving differential equations

The Formal Definition

A function $f(x)$ is continuous at a point $x=a$ if three conditions are met:

1. $f(a)$ exists (the function is defined at $a$)
2. ${\lim }_{x\to a}f(x)$ exists (the limit exists)
3. ${\lim }_{x\to a}f(x)=f(a)$ (the limit equals the function value)

If a function is continuous at every point in its domain, we say it’s continuous everywhere.

Simple Examples

Example 1: A Continuous Function

Take $f(x)=x^2$.

• At $x=2$: $f(2)=4$
• As $x$ approaches 2: $f(x)$ gets closer and closer to 4
• No holes or jumps, so continuous at $x=2$

You can draw $y=x^2$ without lifting your pencil anywhere.

Example 2: A Discontinuous Function

Take $f(x)=\frac{1}{x}$.

• At $x=0$: $f(0)$ is undefined (division by zero)
• As $x$ approaches 0 from the right: $f(x)\to +\infty$
• As $x$ approaches 0 from the left: $f(x)\to -\infty$
• There’s a vertical asymptote at $x=0$, so discontinuous there

The graph has a break at $x=0$.

Example 3: Piecewise Function

Consider:

x & \text{if } x \geq 0 \\ -x & \text{if } x < 0 \end{cases}$$ This is the absolute value function: $f(x) = |x|$. - At $x = 0$: $f(0) = 0$ - From the right: approaches 0 - From the left: approaches 0 - All three conditions are met, so continuous at $x = 0$ The "V" shape is continuous, even though it's not smooth at the point. ## Types of Discontinuities ### 1. Removable Discontinuity A hole in the graph that can be "filled" by redefining the function. Example: $f(x) = \frac{x^2 - 1}{x - 1}$ - Simplifies to $f(x) = x + 1$ for $x \neq 1$ - At $x = 1$, there's a hole, but if we define $f(1) = 2$, it becomes continuous ### 2. Jump Discontinuity The function jumps from one value to another. Example: Step function $$f(x) = \begin{cases} 0 & \text{if } x < 0 \\ 1 & \text{if } x \geq 0 \end{cases}$$ At $x = 0$, left limit is 0, right limit is 1, so discontinuous. ### 3. Infinite Discontinuity The function goes to infinity, like $f(x) = \frac{1}{x}$ at $x = 0$. ## Applications in Game Development In Unity or other engines: - Continuous position functions ensure smooth character movement - Continuous time-based functions prevent jerky animations - Physics engines assume continuous force functions for realistic simulations Continuity ensures that small changes in input lead to small changes in output, which is essential for stable game mechanics.