# Discontinuity of a piecewise defined function with a parameter

Let

$$f(x,y) = \left\{ \begin{array}{ll} cx+4, & \textrm{if x<6,}\\ cx^2-4, & \textrm{if x\geq 6.}\\ \end{array} \right.$$

respectively. For what value of $c$ is this function continuous on $(-\infty, \infty)$. I have tried everything in my mind and I am getting no where, please help.

### Update

I figured it out. I just forced $6c+4=36c-4$ so that the hole from values of $x<6$ would be filled by $x=6$. So you get $4/15$.

• Can you tell us your thoughts? What have you tried? – RghtHndSd Sep 11 '14 at 2:22

## 1 Answer

Hint. The function is continuous at $6$ if and only if $$\lim_{x\to6^-}f(x)\ ,\quad f(6)\ ,\quad \lim_{x\to6^+}f(x)$$ are all the same. See if you can calculate these three quantities (in terms of $c$ where necessary), and then find the value of $c$ for which all three are equal.

• I figured it out guys. I just forced 6c+4=36c-4 so that the hole from values of x<6 would be filled by x=6. So you get 4/15. – Dr. Coffeeisgood Sep 11 '14 at 2:20