(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!

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