Continuity
No breaks, jumps or holes: limit equals function value.
Key idea
No breaks, jumps or holes: limit equals function value.
In depth
A function is continuous at a if f(a) is defined, the limit exists, and the limit equals f(a). Graphically, you can draw it without lifting your pencil. Polynomials, sine and exponentials are continuous everywhere on their domains.
Worked example
f(x) = x² is continuous at 2 since f(2) = 4 and the limit is 4.
Formula
Real-world uses
- Checking that a physical model such as a smooth road profile or a temperature curve has no jumps
Common mistake
Checking only the limit without confirming f(a) is defined.
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 Continuity?
No breaks, jumps or holes: limit equals function value. It is a topic at the advanced level in Calculus.
What is the formula for Continuity?
The key formula is lim f(x) = f(a).
Where is Continuity used in real life?
Real-world examples: Checking that a physical model such as a smooth road profile or a temperature curve has no jumps.
What should I learn before Continuity?
Make sure you are comfortable with Squeeze Theorem, the previous topic in Calculus.
What is a common mistake with Continuity?
Checking only the limit without confirming f(a) is defined.
How can I practise Continuity?
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 Continuity?
Move on to Types of Discontinuity 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.