Definition: A function is continuous at a point when the limit exists there, the function is defined there, and the two values are equal, so its graph has no break.
What does continuous mean?
Informally, you can draw the graph without lifting your pencil. Formally, f is continuous at x = a when three things hold: f(a) exists, the limit as x approaches a exists, and the two are equal.
How do you make a piecewise function continuous?
Match the left and right pieces at the join.
Worked example: f(x) = kx – 5 for x at most 2, and f(x) = x2 + 1 for x greater than 2. The right piece gives 4 + 1 = 5 at x = 2. The left piece gives 2k – 5. Set 2k – 5 = 5, so k = 5.
With k = 5 both pieces meet at the point (2, 5).
What are the types of breaks?
Removable (a hole), jump (sides disagree), and infinite (vertical asymptote). Mathbot notes that continuity is a pointwise idea: a function can be continuous at one point and broken at another.