Construct (by sketching) a function h defined on the closed interval [−4,4] that satisfies all the conditions given below. After sketching, provide justification that each condition is met.
• h is continuous at every point of (−4,4) except at x=−1,2,3.
• At x=−1 the discontinuity is removable.
• At x=2 the discontinuity is a jump discontinuity.
• At x=3 the discontinuity is an infinite (essential) discontinuity.
• h(−4)=0 and h(4)=3, and the function is continuous from the right at x=−4 and from the left at x=4.
a. (i) Draw a clear, labelled sketch of such a function on the interval [−4,4].
b. (ii) Explain, with reference to limits and function values, why your sketch fulfils each of the five stated conditions.