Continuity of a function at a point

What it shows

A function f is continuous at x₀ when the left-hand limit, the right-hand limit and the value f(x₀) all exist and are equal. The simulation uses piecewise functions joined at x₀ whose right branch contains a parameter m, for example x² + 1 for x < x₀ and mx + 2 for x ≥ x₀. It evaluates the three numbers and marks the size of the jump between the branches. In the rational form, (x² − x₀²)/(x − x₀) is undefined at x₀ but reduces to x + x₀, so its limit 2x₀ still exists and a suitable value can close the gap.

How to use

Choose a Function type and a Join point x₀, then drag Parameter m slowly and watch the jump label shrink. Ask students to write the condition that the left-hand limit equals f(x₀) and solve it for m on paper first. They then check their answer with Set m for continuity and confirm that the graph can be drawn without lifting the pen. Finally try x₀ = 0 in the quadratic–linear form, where no value of m works.

Parameters you can change

• Join point x₀ -3–3
• Parameter m -6–6
• Function type Left branch quadratic – right branch linear, Left branch a reducible rational expression – right branch constant

Questions to explore

1. Which three quantities must exist and be equal for f to be continuous at x₀?
2. For f(x) = x² + 1 when x < 1 and mx + 2 when x ≥ 1, which m makes f continuous?
3. Why does (x² − 4)/(x − 2) have a limit at x = 2 even though it is undefined there?