How can I choose a constant to make a piecewise function continuous?

At the joining point, make the value approached from the left equal the value approached from the right, then make that shared value equal the function’s actual value at the point. Continuity requires the function to be defined there, its two-sided limit to exist, and that limit to equal the function value. [S1]

For example, suppose

f(x) = x + 1 when x < 2, and f(x) = x² + c when x ≥ 2.

The joining point is x = 2. The left-hand rule gives a limit of 2 + 1 = 3. The right-hand rule also defines f(2), because its condition includes x = 2; that value is 2² + c = 4 + c. Set them equal: 3 = 4 + c, so c = -1. The rules then meet at 3.

Because the inequalities tell you which piece actually includes the joining point. In this example, x ≥ 2 includes x = 2, so f(2) is calculated with x² + c. The left piece, x + 1, applies only when x < 2; it tells you what the function approaches from the left, not what its assigned value is at 2.

Keep the three checks distinct:

1. Left-hand limit: use the formula for x < 2.
2. Right-hand limit: use the formula for x > 2 (here, the same expression as the x ≥ 2 piece).
3. Function value: use the piece whose condition includes x = 2.

Here the right-hand limit and f(2) are both 4 + c. Setting that equal to the left-hand limit, 3, satisfies all three checks.

Substitute c = -1 into the piecewise definition:

f(x) = x + 1 when x < 2, and f(x) = x² - 1 when x ≥ 2.

From the left, as x approaches 2, x + 1 approaches 3. From the right, x² - 1 approaches 2² - 1 = 3. At the point itself, the second piece applies, so f(2) = 2² - 1 = 3. The two-sided limit exists, and it equals f(2); therefore the function is continuous at x = 2. [S1]

Write all three results explicitly. If the left and right limits differ, the two-sided limit does not exist, so changing a constant in the point’s assigned value alone cannot make the function continuous.

Let the joining point be x = a. Identify which formula applies for x < a, which applies for x > a, and which piece includes x = a. Call the unknown constant c. Then calculate L, the left-hand limit; R, the right-hand limit; and V = f(a). For continuity, solve L = R and then make their common value equal V. If the point belongs to the left piece instead, use that piece to calculate V; the method does not depend on the right piece owning the endpoint.

Copyable setup:

Joining point: a = [point] Left-hand limit: L = [evaluate left formula as x approaches a] Right-hand limit: R = [evaluate right formula as x approaches a] Function value: V = [evaluate the piece containing a at x = a] Continuity equations: L = R and R = V Solution: c = [solved value] Check: L = R = V = [common value]

This method checks continuity at that joining point. It does not by itself establish continuity at other points in the function’s domain.