Answer
See below for answers and explanations.
Work Step by Step
a. Based on the given piece-wise function, we have
$\lim_{x\to8500^-}T(x)=0.1(8500)=850$
and
$\lim_{x\to8500^+}T(x)=850+0.15(8500-8500)=850$
Since $T(x)$ is defined at $x=8500$ and the left and right limits exist and are equal to the function value, we conclude that $T$ is continuous at $8500$.
b. We have
$\lim_{x\to34500^-}T(x)=850+0.15(34500-8500)=4750$
and
$\lim_{x\to34500^+}T(x)=4750+0.25(34500-34500)=4750$
Since $T(x)$ is defined at $x=34500$ and the left and right limits exist and are equal to the function value, we conclude that $T$ is continuous at $34500$.
c. If $T$ contains discontinuities, it is possible that a person with a higher earning pays too much tax and that he ends up getting less money in his pocket than another person with a lower earning who pays less tax.
