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 $x=1$, we need to evaluate the left and right limits at this point and compare with the function value. We have
$\lim_{x\to1^-}x=1$, $\lim_{x\to1^+}(2-x)=1$, and $f(1)=1$. Since these values are equal, we conclude that the function is continuous at $x=1$.
c. To see if the function is differentiable at $x=1$, we need to evaluate the left and right derivatives at this point. We have $\lim_{x\to1^-}f'(x)=\lim_{x\to1^-}1=1$ and $\lim_{x\to1^+}f'(x)=\lim_{x\to1^+}(-1)=-1$. Since these values are not equal, we conclude that the function is not differentiable at $x=1$.
