Question

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

T(x) = (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.

Answer 1

(i) To show that a piecewise function is continuous at a point, we need to show that the left hand and right hand limit "agree" with each other. In other words, we want:

$\lim_{x\to 6061^-} T(x) = \lim_{x \to 6061^+} T(x)$

Now, since we're given the constraints and the equation of each constraint, we notice that 6061^+ is a number that is slightly bigger than 6061. So we use the second equation. Do you see why?

In much the same way, 6061^- is a number that is slightly smaller than 6061. So we use the first equation. Again, do you see why? (Hint: look at the conditions on x for each equation).

So finally, computing each limit means just "plugging" 6061 into their respective equations. That is:

$\lim_{x \to 6061^-} T(x) = 0.10\times 6061 = 606.1$

$\lim_{x \to 6061^+} T(x) = 606.1 + 0.18(6061 - 6061) = 606.1$

Since your limits match, we say that, at the point x = 6061, T(x) IS continuous.

(ii) Repeat the process above with x = 32473.

(iii) Find a point of discontinuity just means your right hand and left hand limits do not match -- I'm not an economist, so I may not be of much help with the latter part of the question!

Answer 2

Answer with explanation:

For a piece-wise function to be continuous we need to only check the function at the nodes i.e. at the starting and end points.

a)

The function T(x) is given by:

T(x)=    0.10 x       if        0<x≤6061

          606.10+0.18(x-6061)  if  6061 <x≤32473

Now to check whether T(x) is continuous at  x=6061 we need to check the left and right hand limit of the function.

Left hand limit at x=6061 is:

lim x→6061    0.10x

      = 6061.10

Also, the right hand limit of function at x=-6061 is:

lim x→6061   606.10+0.18(x-6061)

            = 606.10+.18(6061-6061)

           = 606.10

Hence, the left hand and right hand limit of the function is equal and equal to the value of the function at x=6061

Hence, the function T(x) is continuous at x=6061

b)

Now we have to check that T(x) is continuous at x=32473

The function T(x) is defined by:

  T(x)=   606.10+0.18(x-6061)  if  6061 <x≤32473

 5360.26+0.26(x-32473)     if  32473<x≤72784

Left hand limit at x=32473 is:

lim x→32473   606.10+0.18(x-6061)

= 606.10+0.18(32473-6061)

             

Hence, left hand limit equal to right hand limit is equal to function value at x=32473

Hence, the function T(x) is continuous at x=32473.

c)

Similarly when we will check at the other nodal points we get that the function is continuous everywhere in the given domain.