Concept preview

What you will be able to do

1. 01State the three conditions that must all hold for a function to be continuous at a point, and explain what each one rules out.
2. 02Test a given function (including piecewise and rational functions) for continuity at a specific x-value by checking value, limit, and agreement.
3. 03Determine the parameter value that makes a piecewise function continuous at a junction by forcing the one-sided limits and the function value to agree.

In earlier lessons you learned to compute a limit: the single height a graph approaches as $x$ closes in on a target from both sides. Now we connect that idea back to the function itself. A limit describes where the graph is heading; the function value $f(c)$ describes where the graph actually is. Continuity is the study of when those two agree. When they do, the graph flows through the point smoothly; when they do not, there is a hole, a jump, or a blow-up. This single comparison is one of the most important ideas in all of calculus, because every major theorem ahead — the Intermediate Value Theorem, the Extreme Value Theorem, differentiability, and the definite integral — quietly assumes continuity.