On the interval [-2,6], the graph of a function r consists of three line segments: the…

On the interval \([-2,6]\), the graph of a function \(r\) consists of three line segments: the segment joining \((-2,5)\) to \((0,-1)\), the segment joining \((0,2)\) to \((3,2)\), and the segment joining \((3,-4)\) to \((6,2)\). At each jump, the graph includes the left-hand point and excludes the right-hand copy of that input, so \(r(0)=-1\) and \(r(3)=2\). The graph includes the endpoints \((-2,5)\) and \((6,2)\). (a) Find \(r(-1)\), \(r(0)\), \(r(2)\), \(r(3)\), 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 \(\lim_{x\to 3^-}r(x)\) and \(\lim_{x\to 3^+}r(x)\). Is \(r\) continuous at \(x=3\)? Use the definition of continuity. (d) List every value of \(x\) in the open interval \((-2,6)\) at which \(r\) is discontinuous. (e) Find the average rate of change of \(r\) on the interval \([-2,0]\).

Piecewise linear graphs are ordinary lines between the given vertices; the jump rule only decides which value is actually attained at a break.