Answer
See graph and explanations.
Work Step by Step
a. See graph for the piecewise function.
b. To see if the function is continuous at $x=0$, we need to evaluate the left and right limits at this point and compare with the function value. We have
$\lim_{x\to0^-}x^2=0$, $\lim_{x\to0^+}(-x^2)=0$, and $f(0)=0$. Since these values are equal, we conclude that the function is continuous at $x=0$.
c. To see if the function is differentiable at $x=0$, we need to evaluate the left and right derivatives at this point. We have $\lim_{x\to0^-}f'(x)=\lim_{x\to0^-}(2x)=0$ and $\lim_{x\to0^+}f'(x)=\lim_{x\to0^+}(-2x)=0$. Since these values are equal, we conclude that the function is differentiable at $x=0$.
