Types of Discontinuity
Removable, jump and infinite discontinuities.
Key idea
Removable, jump and infinite discontinuities.
In depth
Removable: the limit exists but differs from or is missing at f(a). Jump: the one-sided limits differ. Infinite: the function blows up. Oscillating: it fails to settle. Removable ones can be repaired by redefining one value.
Worked example
f(x) = (x² − 1)/(x − 1) has a removable hole at x = 1 with limit 2.
Real-world uses
- Spotting jumps in tax tables, holes in data sets and asymptotes in models
Common mistake
Calling a jump removable.
How to learn this
- Read the key idea and say it in your own words.
- Work through the worked example, then try a similar problem.
- Use the interactive lesson’s visual and mini-quiz to check the pattern.
Frequently asked questions
What is Types of Discontinuity?
Removable, jump and infinite discontinuities. It is a topic at the advanced level in Calculus.
Where is Types of Discontinuity used in real life?
Real-world examples: Spotting jumps in tax tables, holes in data sets and asymptotes in models.
What should I learn before Types of Discontinuity?
Make sure you are comfortable with Continuity, the previous topic in Calculus.
What is a common mistake with Types of Discontinuity?
Calling a jump removable.
How can I practise Types of Discontinuity?
Open the interactive lesson, play with the visual, then answer the three-question mini-quiz. Getting two right marks the topic complete.
What comes after Types of Discontinuity?
Move on to Intermediate Value Theorem next.
Related advanced topics in Calculus
- Idea of a Limit – The value a function approaches as the input approaches a point.
- One-Sided Limits – Left and right limits must agree for the limit to exist.
- Limit Laws – Limits of sums, products and quotients are computed from the parts.
- Factoring and Rationalizing Limits – Remove 0/0 indeterminacy with algebra.
- Limits at Infinity – End behaviour of functions.
- Infinite Limits and Asymptotes – Function values grow without bound near a vertical asymptote.