Prerequisites
Evaluate a linear function and read open and filled coordinate points.
Read a left-hand limit using inputs below the target, and a right-hand limit using inputs above it. A two-sided finite limit exists only when both approach the same finite number. A point value cannot repair disagreeing sides. A short table suggests behavior but does not define every nearby output.
Worked example
For x other than -1, a separate function R follows R(x) = 3x + 2, while R(-1) = 5. Both sides approach -1, not 5. The nearby expression approaches 3(-1) + 2 = -1. To make R continuous there, give R(-1) that same value.
Three new function records
P: For every x other than 3, P(x) = 2x - 1. At exactly 3, P(3) = 8. The graph has an open circle at (3, 5) and a filled point (3, 8). Q: For x < 1, Q(x) = 4. For x > 1, Q(x) = 0. At exactly 1, Q(1) = 2. R: Only four readings are supplied: R(-0.1) = 6.9, R(-0.01) = 6.99, R(0.01) = 7.01, R(0.1) = 7.1. Values at other nearby inputs are unspecified.