Continuity and Discontinuity
Lesson notes
Overview
A function is continuous if you can draw it without lifting the pencil. This topic is rated intermediate and is number 65 of 245 in Mathematics. Builds on the basics with relationships, methods and trade-offs.
What you are seeing
The panel above shows a graph of the function. Each slider changes one parameter, so you can see exactly how the curve bends, shifts or stretches. The animation is drawn live from the model, so everything you see responds to the sliders. Press Space to pause, use Restart to replay from the start, and change the speed to slow down quick events.
In depth
A function is continuous at a point if its limit there exists and equals its value, which in simple terms means you can draw it without lifting the pencil. Discontinuities come in kinds: a removable hole, a jump between two different values, or an infinite asymptote. Continuous functions have useful properties, such as the intermediate value theorem, which guarantees that a continuous function going from negative to positive must cross zero somewhere.
Reading the relationship
The expression "lim f(x) = f(a)" is a compact summary of the idea. It uses the symbols l, i, m, f, x, a; name what each one stands for before you start. Then use the sliders to see how changing one quantity changes the others.
A guided exploration
1. Set every slider to its middle value and note the shape of the curve. 2. Move one slider at a time and describe whether the curve shifts, stretches or turns. 3. Find the value where the behaviour changes sharply, then connect it to the formula.
Try this
Predict the outcome first, then test extreme slider values to find where the pattern changes or breaks.
Common pitfalls
A frequent slip is memorising the rule without the picture. If a result surprises you, sketch the graph or test a simple number by hand before trusting the algebra.
How it connects
It builds on “Limits”, the topic before it in Mathematics. Once you are comfortable, continue with “L’Hôpital’s Rule”.
Why it matters
Mathematical ideas like this one are the language that physics, engineering, computing and statistics are written in.
Where you might use it
- Revising “Continuity and Discontinuity” before a class, exam or certification
- Checking working on homework and exam questions by watching the graph respond
- Modelling growth, change and uncertainty in finance, data science and statistics
- Building the reasoning behind the algorithms and engineering tools people use every day
Frequently asked questions
What is Continuity and Discontinuity?
A function is continuous if you can draw it without lifting the pencil. It is an intermediate topic in Mathematics.
How do I use the Continuity and Discontinuity simulation?
The panel at the top shows a graph of the function. Each slider changes one parameter, so you can see exactly how the curve bends, shifts or stretches. Move the sliders one at a time, then press Space to pause the animation and look closer.
What does the formula mean?
The expression lim f(x) = f(a) packs the idea into symbols. Name each symbol first, then change one quantity with the sliders and see which others respond.
Who is this topic for?
Builds on the basics with relationships, methods and trade-offs. If the quiz feels hard, reread the lesson notes and try the previous topic first.
What should I learn before this?
Start with “Limits”, the previous topic in Mathematics. Each topic in a subject builds on the ones before it.
How do I track my progress?
Press “Mark as learned” on this page. Progress is stored only in this browser and is never sent anywhere. The Browse page can show topics you have learned or still need to learn.
Quick quiz
More in Mathematics
See all →L’Hôpital’s Rule
When a ratio is 0/0, compare the rates of change of numerator and denominator.
Optimisation
Calculus finds the best value: the peak of the area of a fence with fixed perimeter.
Critical Points and Concavity
Where the derivative is zero the graph has a flat tangent: a peak, valley or inflection.