Single Variable Calculus, Chapter 2, 2.5, Section 2.5, Problem 41

Determine the value of the constant $c$ that makes the function $f(x) = \left\{ \begin{array}{c} cx^2 + 2x & \text{ if } & x < 2 \\ x^3 - cx & \text{ if } & x \geq 2 \end{array} \right.$ continuous on $(-\infty, \infty)$

Based from the definition of continuity,
The function is continuous of at a number if and only if the left and right hand limits of the function at the same number is equal. So,


$\begin{equation} \begin{aligned} \lim \limits_{x \to 2^-} cx^2 + 2x & = \lim \limits_{x \to 2^+} x^3 - cx\\ c(2)^2+2(2) & = (2)^3 - c(2)\\ 4c + 4 & = 8-2c\\ 4c+2c & = 8-4\\ 6c & = 4\\ c & = \frac{4}{6} = \frac{2}{3} \end{aligned} \end{equation}$


Therefore,
The value of $c$ that would make the function continuous on $(-\infty, \infty)$ is $\displaystyle \frac{2}{3}$