Chapter 11.3, Problem 45PE

The following piecewise function gives the tax owed, T(x), by a single taxpayer for a recent year on a taxable income of x dollars.

$T(x)\{\begin{array}{l}\text{0}\text{.10}x\text{ if 0 < }x\le \text{ 8500}\\ \text{850}\text{.00 + 0}\text{.15(}x\text{ - 8500) if 8500 < }x\le \text{ 34,500}\\ \text{4750}\text{.00 + 0}\text{.25(}x\text{ - 34,500) if 34,500 < }x\le \text{ 83,600}\\ \text{17,025}\text{.00 + 0}\text{.28(}x\text{ - 83, 600) if 83,600 < }x\le \text{ 174,400}\\ \text{42,449}\text{.00 + 0}\text{.33(}x\text{ - 174,400) if 174,400 < }x\le \text{ 379,150}\\ \text{110,016}\text{.50 + 0}\text{.35(}x\text{ - 379,150) if }x\text{ > 379,150}\text{.}\end{array}$

Determine whether T is continuous at 8500.

Determine whether T is continuous at 34,500.

If T had discontinuities, use one of these discontinuities to describe a situation where it might be advantageous to earn less money in taxable income.