Methods for Determining Points of Discontinuity

Determining the type of a point of discontinuity is an important skill in the theory of continuity. This article introduces several common determination methods.

Definition Method

Apply the definition of a point of discontinuity directly:

Steps

1. Check whether the function is defined at that point

   • If the function is undefined at that point, the point may be a point of discontinuity

2. Compute the limit of the function at that point

   • Compute $\lim_{x \to x_0} f(x)$
   • If the limit does not exist, the point is a point of discontinuity

3. Compare the limit with the function value

   • If $\lim_{x \to x_0} f(x) \neq f(x_0)$, the point is a point of discontinuity

Example

Determine the type of discontinuity of $f(x) = \frac{x^2 - 4}{x - 2}$ at $x = 2$

Steps:

1. The function is undefined at $x = 2$
2. Compute the limit: $\lim_{x \to 2} \frac{x^2 - 4}{x - 2} = \lim_{x \to 2} (x + 2) = 4$
3. The limit exists but the function value is undefined
4. Conclusion: $x = 2$ is a removable discontinuity

Left-Right Limit Method

For piecewise functions or complicated functions, the left-right limit method is more convenient:

Steps

1. Compute the left limit and the right limit separately

   • Left limit: $\lim_{x \to x_0^-} f(x)$
   • Right limit: $\lim_{x \to x_0^+} f(x)$

2. Compare the left limit, the right limit and the function value

   • If left limit = right limit = function value, the function is continuous
   • If left limit = right limit ≠ function value, it is a removable discontinuity
   • If left limit ≠ right limit, it is a jump discontinuity
   • If at least one one-sided limit does not exist, it is a discontinuity of the second kind

Example

Determine the type of discontinuity of $f(x) = \begin{cases} x^2, & x < 1 \\ 2x, & x \geq 1 \end{cases}$ at $x = 1$

Steps:

1. Left limit: $\lim_{x \to 1^-} f(x) = \lim_{x \to 1^-} x^2 = 1$
2. Right limit: $\lim_{x \to 1^+} f(x) = \lim_{x \to 1^+} 2x = 2$
3. Function value: $f(1) = 2 \times 1 = 2$
4. Left limit ≠ right limit
5. Conclusion: $x = 1$ is a jump discontinuity

Graphical Method

Determine the type of a point of discontinuity by observing the graph of the function:

Characteristics

1. Continuous point

   • The graph passes through the point continuously
   • There is no break or jump

2. Removable discontinuity

   • The graph has a “hole”
   • The function can be made continuous by filling in the “hole”

3. Jump discontinuity

   • The graph has a “jump”
   • The function values on the two sides are not equal

4. Discontinuity of the second kind

   • The graph may have a vertical asymptote
   • The function values tend to infinity

Example

Determine the type of discontinuity of $f(x) = \frac{1}{x}$ at $x = 0$

Observing the graph:

• The graph of the function has a vertical asymptote at $x = 0$
• It tends to negative infinity on the left and to positive infinity on the right
• Conclusion: $x = 0$ is a discontinuity of the second kind

Flowchart for Determining a Point of Discontinuity

Start
  ↓
Is f(x) defined at x₀?
  ↓ No → Compute the limit
  ↓ Yes
  ↓
Compute lim f(x)
  x→x₀
  ↓
Does the limit exist?
  ↓ No → Compute the left-hand and right-hand limits
  ↓ Yes
  ↓
Is the limit = f(x₀)?
  ↓ No → Removable discontinuity
  ↓ Yes
  ↓
Continuous point

Exercises

Exercise 1

Use the definition method to determine the type of discontinuity of $f(x) = \frac{x^2 - 9}{x - 3}$ at $x = 3$.

Exercise 2

Use the left-right limit method to determine the type of discontinuity of $f(x) = \begin{cases} x, & x < 0 \\ x + 1, & x \geq 0 \end{cases}$ at $x = 0$.

Exercise 3

Use the graphical method to determine the type of discontinuity of $f(x) = \frac{1}{x^2}$ at $x = 0$.

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Summary

Symbols Used in This Article