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!
Best Answer
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 $3$.