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Continuity at a joint

A function is continuous at a point when the limit there equals the function value. The graph can then be drawn without lifting the pencil. For a piecewise formula, the value from the left and the value from the right must be equal at the joint. You set the two expressions equal and solve for the unknown constant. Away from the joint, each polynomial piece is already continuous.

\lim_{x \to a} f(x) = f(a)

continuity

f(a^-) = f(a^+)

the joint matches

Symbols

$a$	the joint
$k$	the constant you must find

Example

$f(x) = x^2 + k$ for $x < 1$ and $f(x) = 2x + 3$ for $x \ge 1$ . At $x = 1$ the right-hand piece is $5$ , so $1 + k = 5$ and $k = 4$ .

Both pieces must give the same number at the joint. It is not enough that each piece is defined.

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Part of Mathematics R1: Limits and continuity.