Continuity

Definition of Continuity

Continuity refers to a function being "smooth" without any breaks, jumps, or holes at a given point or over an interval. Mathematically, a function $f(x)$ is continuous at a point $c$ if:

1. The function is defined at $c$, i.e., $f(c)$ exists.
2. The limit $\lim_{x \to c} f(x)$ exists.
3. The value of the limit equals the function value, i.e., $\lim_{x \to c} f(x) = f(c)$.

If any of these conditions fail, the function is said to be discontinuous at $c$.

Types of Discontinuities

Discontinuities occur when a function fails to be continuous at a point. The three main types of discontinuities are:

• Removable Discontinuity: A "hole" in the graph that can be "fixed" by redefining the function at that point.
• Jump Discontinuity: The function has a sudden "jump" where the left-hand and right-hand limits are not equal.
• Infinite Discontinuity: The function approaches infinity near a point, often due to division by zero.

Continuity Over an Interval

A function is continuous over an interval if it is continuous at every point within that interval. For example:

• Open Interval: Continuous on $(a, b)$ means the function is continuous for all points between $a$ and $b$.
• Closed Interval: Continuous on $[a, b]$ means it is continuous on the open interval $(a, b)$ and at the endpoints $a$ and $b$.

Example Question

Problem:

Determine whether the function $f(x) = \frac{x^2 - 1}{x - 1}$ is continuous at $x=1$.

Solution:

1. The function $f(x)$ is undefined at $x=1$ because the denominator $x-1$ becomes zero.
2. To check for continuity, simplify the function:
   $$f(x) = \frac{(x-1)(x+1)}{x-1}$$
   For $x \neq 1$, the function simplifies to:
   $$f(x) = x + 1$$
3. Find the limit of $f(x)$ as $x \to 1$:
   $$\lim_{x \to 1} f(x) = 1 + 1 = 2$$
4. However, $f(x)$ is not defined at $x=1$, so the function has a **removable discontinuity** at $x=1$.

Practical Applications of Continuity

Continuity is a vital concept in various fields:

• Physics: Ensures smooth transitions in motion or energy systems.
• Engineering: Models continuous systems like signal processing.
• Economics: Analyzes cost functions without abrupt changes.
• Computer Science: Ensures seamless animations and transitions in graphics.

Practice Question

Problem:

Verify the continuity of the function $f(x) = \begin{cases} x^2 & \text{if } x \leq 2 \\ 3x - 4 & \text{if } x > 2 \end{cases}$ at $x=2$.

Solution:

1. Check if $f(2)$ is defined:
   $$f(2) = 2^2 = 4$$
2. Calculate the left-hand limit:
   $$\lim_{x \to 2^-} f(x) = \lim_{x \to 2^-} x^2 = 4$$
3. Calculate the right-hand limit:
   $$\lim_{x \to 2^+} f(x) = \lim_{x \to 2^+} (3x - 4) = 3(2) - 4 = 2$$
4. Since the left-hand limit and right-hand limit are not equal, the function is **not continuous** at $x=2$.