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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! |
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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 $3$. |
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