2.4 Making a Function Continuous
Notes
To make a piecewise function continuous at a boundary, make the left and right limits equal to each other and to the function value.
- Set the two pieces equal at the boundary and solve for the unknown constant.
- Example: $f(x) = 3x + 2$ for $x < 1$ and $f(x) = kx - 1$ for $x \geq 1$. The left side gives 5, so $k - 1 = 5$ and $k = 6$.
- For a hole, define the missing value as the limit.
Practice
Let $f(x) = 3x + 2$ for $x < 1$ and $f(x) = kx - 1$ for $x \geq 1$. For what $k$ is $f$ continuous at $x = 1$?
The left limit is $3(1) + 2 = 5$. Set $k(1) - 1 = 5$.
Let $\displaystyle f(x) = \frac{x^2 - 16}{x - 4}$ for $x \neq 4$ and $f(4) = k$. For what $k$ is $f$ continuous at 4?
For $x \neq 4$, $f(x) = x + 4$, so the limit at 4 is 8. Set $k = 8$.
Let $f(x) = x^2 + a$ for $x \leq 2$ and $f(x) = 2x + 7$ for $x > 2$. For what $a$ is $f$ continuous at 2?
The right side gives $2(2) + 7 = 11$. Set $4 + a = 11$.
Ready for more? Take the Unit 2 test.