(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:
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:
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!