1
$\begingroup$

Working on my AP Calc summer assignment and I am having a hard time understanding how to solve this; I could really use some very dumbed-down help if possible because I don't even know where to start. Here it is...

Determine where the function is continuous and where the function is differentiable.

$f(x)=\begin{cases}(x+1)^2,& x \leq 0\\ 2x+1,& 0< x < 3\\ (4-x)^2,& x \geq 3\end{cases}$

Thank you in advance for your help!

  • 0
    That makes sense but how would I go about doing that? I've never worked with piecewise functions before.2011-08-23

1 Answers 1

2

As is said in the comments, everything is clear except at $0$ and $3$.

To see if it is continuous at $0$, for example, you need to check that the definition of continuity at a point is satisfied at $0$. That is, is it true that $\displaystyle \lim_{x\to 0}f(x) = f(0)$?

For differentiability, you again need to check the definition: Does the limit $\displaystyle \lim_{h \to 0} \frac{f(0+h)-f(0)}{h}$ exist?

Similar checks need to be performed for behavior at $3$.

  • 3
    @Kaleidoscopic if you find an answer to your question useful, then it is good practice to accept an answer. This will serve to let the community know that your question has been resolved.2011-08-24