On the interval [-4,6], the graph of a function r consists of three line segments: the…

On the interval \([-4,6]\), the graph of a function \(r\) consists of three line segments: the segment joining \((-4,3)\) to \((0,-1)\), the segment joining \((0,2)\) to \((4,2)\), and the segment joining \((4,-3)\) to \((6,1)\). At each jump, the graph includes the left-hand point and excludes the right-hand copy of that input, so \(r(0)=-1\) and \(r(4)=2\). The graph includes the endpoints \((-4,3)\) and \((6,1)\). (a) Find \(r(-2)\), \(r(0)\), \(r(2)\), \(r(4)\), and \(r(6)\). (b) Evaluate \(\lim_{x\to 0^-}r(x)\) and \(\lim_{x\to 0^+}r(x)\). Does \(\lim_{x\to 0}r(x)\) exist? (c) Evaluate \(\lim_{x\to 4^-}r(x)\) and \(\lim_{x\to 4^+}r(x)\). Is \(r\) continuous at \(x=4\)? Use the definition of continuity. (d) List every value of \(x\) in the open interval \((-4,6)\) at which \(r\) is discontinuous. (e) Find the average rate of change of \(r\) on the interval \([-4,0]\).

Each piece is a line segment. The jump rule includes the left-hand endpoint, so \(r(0)=-1\) and \(r(4)=2\). Continuity at a point requires that the two-sided limit exist and equal the function value.