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.

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lidaralbany lidaralbany · Answer 1 · 2019-06-24T12:26:13+02:00

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.