Continuity and Discontinuity of Functions

Types of Discontinuities

In calculus, understanding the behavior of functions at specific points is crucial. Discontinuities are points where a function is not continuous. There are two main types:

• Removable Discontinuity: Occurs when a function has a 'hole' at a point, but the limit exists. The discontinuity can be 'fixed' by redefining the function at that point.

• Nonremovable Discontinuity: Includes jump and infinite discontinuities. These cannot be fixed by redefining the function at a single point.

Examples:

• Jump discontinuity: $f(x)$ has different left and right limits at $x=c$.

• Infinite discontinuity: $f(x)$ approaches infinity as $x$ approaches $c$.

• Removable discontinuity: $f(x)$ is undefined at $x=c$, but $\lim_{x \to c} f(x)$ exists.

Definition of Continuity at a Point

A function $f$ is continuous at a point $x=c$ if all the following conditions are met:

• I. $f(c)$ exists: The function is defined at $x=c$.

• II. $\lim_{x \to c} f(x)$ exists: The left and right limits as $x$ approaches $c$ are equal and finite.

• III. $\lim_{x \to c} f(x) = f(c)$: The value of the function at $c$ matches the limit.

Formula:

• $\lim_{x \to c^-} f(x) = \lim_{x \to c^+} f(x) = f(c)$

Discontinuity at a Point

If any of the above conditions fail, the function is discontinuous at $x=c$. The graph of $f$ will show a break, jump, or hole at $c$.

• Type 1: $f(c)$ is not defined.

• Type 2: $\lim_{x \to c} f(x)$ does not exist.

• Type 3: $\lim_{x \to c} f(x) \neq f(c)$.

Example: Using the Continuity Test

To determine continuity at $x=2$ for $f(x)$:

• Check if $f(2)$ exists.

• Compute $\lim_{x \to 2^-} f(x)$ and $\lim_{x \to 2^+} f(x)$.

• If both limits exist and are equal to $f(2)$, the function is continuous at $x=2$.

Example Calculation:

• If $f(x) = x^2$ for $x < 2$, $f(x) = 2x$ for $x \geq 2$, then $\lim_{x \to 2^-} f(x) = 4$, $\lim_{x \to 2^+} f(x) = 4$, $f(2) = 4$ → continuous at $x=2$.

Determining Values for Continuity

Sometimes, a parameter must be chosen so that a piecewise function is continuous everywhere.

• Set $\lim_{x \to c^-} f(x) = \lim_{x \to c^+} f(x)$ and solve for the parameter.

• Example: For $f(x) = \begin{cases} x^2 + c, & x < 2 \\ 2x + 5, & x \geq 2 \end{cases}$, set $\lim_{x \to 2^-} f(x) = \lim_{x \to 2^+} f(x)$ to solve for $c$.

Calculation:

• $\lim_{x \to 2^-} f(x) = 4 + c$, $\lim_{x \to 2^+} f(x) = 9$ → $4 + c = 9$ → $c = 5$

Theorem: Properties of Continuous Functions

If $f$ and $g$ are continuous at $x=c$, then the following combinations are also continuous at $x=c$:

Practice: Identifying Discontinuities on a Graph

Given a graph, identify:

• The $x$-values where discontinuities occur

• Which step of the continuity test is broken

• The type of discontinuity (removable, jump, infinite)

Intermediate Value Theorem (IVT)

Statement of IVT

The Intermediate Value Theorem states: If $f$ is continuous on $[a, b]$ and $k$ is any value between $f(a)$ and $f(b)$, then there exists at least one $c$ in $[a, b]$ such that $f(c) = k$.

• Application: Used to show that a function has a root (zero) in an interval.

Example:

• Show $f(x) = x^2 - 2 - \cos x$ has a zero in $[0, \pi]$:

• $f(0) = -1$, $f(\pi) = \pi^2 - 2 + 1 = \pi^2 - 1$

• Since $f(0) < 0$ and $f(\pi) > 0$, IVT guarantees a $c$ in $[0, \pi]$ such that $f(c) = 0$.

Using IVT to Solve for $c$

Given $f(x)$ and an interval $[a, b]$, check:

• Is $f$ continuous on $[a, b]$?

• Does $k$ lie between $f(a)$ and $f(b)$?

• If yes, IVT applies and there exists $c$ such that $f(c) = k$.

Example:

• For $f(x) = \frac{x^2 - x}{x^2 - 5x + 6}$, find $c$ such that $f(c) = 0$ on $[2.5, 4]$.

• Check continuity and solve $x^2 - x = 0$ → $x=0$ or $x=1$ (must be in interval).

Summary Table: Continuity Test and Discontinuity Types

Additional info:

• Continuity is a foundational concept for calculus, essential for understanding limits, derivatives, and integrals.

• Piecewise functions often require careful analysis at the points where the formula changes.

• The IVT is a key tool for proving the existence of solutions to equations within intervals.