# How to determine where the function is continuous and where the function is differentiable with a piecewise function

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.

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

• Hint: Is it already clear to you that the function is continuous and differentiable everywhere except possibly 0 and 3, and you simply need to find which properties it fulfills at 0 and 3? – cobaltduck Aug 23 '11 at 17:29
• That makes sense but how would I go about doing that? I've never worked with piecewise functions before. – Kaleidoscopic Aug 23 '11 at 17:39

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 $3$.