Which one of the following functions is continuous at x = 3?

Concept

A function f is continuous at a point x = a only when all three of these exist and are equal: the value f(a), the left-hand limit (LHL) as x → a⁻, and the right-hand limit (RHL) as x → a⁺. If even one is missing or unequal, f is discontinuous there.

Application at x = 3

Evaluate the three quantities for the piecewise rule that uses (x + 3)/3 below 3, x − 1 above 3, and the value 2 at the point itself:

1. LHL: as x → 3⁻ use (x + 3)/3, giving (3 + 3)/3 = 2.

2. RHL: as x → 3⁺ use x − 1, giving 3 − 1 = 2.

3. Point value: f(3) = 2 is defined directly.

Cross-check

All three agree: LHL = RHL = f(3) = 2, so the three-part test is satisfied and this rule is continuous at x = 3. The others fail it — a fixed point value that disagrees with the common limit, a rule that never assigns a value at x = 3, and a rule whose denominator collapses to 0 at x = 3.

Result: the continuous function is the one defined by (x + 3)/3 for x < 3, x − 1 for x > 3, and 2 at x = 3.