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

Screenshot of the function: http://prntscr.com/14yki3

(i) Determine whether T is continuous at 6061.
(ii) Determine whether T is continuous at 32,473.
(iii) 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.

Question

The following piecewise function gives the tax owed, T(x), by a single taxpayer on a taxable income of x dollars. 

Screenshot of the function: http://prntscr.com/14yki3

 (i) Determine whether T is continuous at 6061. 
(ii) Determine whether T is continuous at 32,473.
 (iii) 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.

anonymous · Answer

For $T(x)$ to be continuous at $6,061$ and $32,473$, the following must be satisfied:
$$(i)~~~\lim_{x\to6,061^-}T(x)=\lim_{x\to6,061^+}T(x),~\text{and }T(6,061)\text{ must exist}\
(ii)~~~\lim_{x\to32,473^-}T(x)=\lim_{x\to32,473^+}T(x),~\text{and }T(32,473)\text{ must exist}$$

anonymous · Answer

I'm getting Math Processing Errors. Sorry.

anonymous · Answer

So you're having a problem with taking the actual limits, or is that some sort of error displayed on a calculator?

anonymous · Answer

Oh, no, the mathy bits you wrote are coming up as [Math Processing Error].

anonymous · Answer

Oh. Basically, i just wrote down the requirements for T(x) to be continuous at 6061 and 32473.

You must have the one-sided limits (from both sides) equal to each other, and T(6061) and T(32473) must exist.|dw:1368632340557:dw|