Explain why each function is continous or discountinous?
(a). The temp. at a specific location as a function of time?
(b). The temp. at a specific time as a function of the distance due west from New York City?
(c). the altitute above sea level as a function of the distance due west from new York Cty?
(d). the cost of a taxi ride as a function of the distance traveled?
(e). The current in the circuit for the lights in a room as function of time?
A continuous function takes on every value between any two given outputs. ( Thus if a<b and f(a)<f(b), then for f(a)<k<f(b) there exists a<c<b such that f(c)=k. The condition holds if you reverse the inequalities.) Discrete functions jump from one output value to another abruptly.
Two other forms of discontinuity include a "hole" in the graph and if the function increases/decreases without bound at a point.
(a) Temperature takes on all values between two values so this is continuous. (e.g. if the temperature at 12:00pm is 50 and the temperature at 3:00pm is 75, then there is at least one point in time from 12-3 where the temperature is 68.7937 degrees, or any other degree you name between 50 and 75.)
(b) I would argue that this function is also continuous. As you move through a distance, the temperature might be changing, even rapidly (think suddenly entering the shadow of a hill/mountain etc...) but the air will flow from warm to cold so there will still be a continuous change.
It is equally plausible to argue that the abrupt change in temperature occurs without any gradation. If so, the function is not continuous.
(c) I would argue that this might not be continuous as you might encounter a sheer cliff.
(d) This is not continuous. There is typically an initial charge, then you are charged every 1/10 mile, or other increment. Thus the charge only changes, in my example, every 1/10 of a mile, then abruptly increases without going through the other costs.
(e) Probably not continuous as you can shut the lights off or turn them on, thus creating a jump discontinuity. I don't know enough physics to argue whether the current flows "instantaneously" (hampered by the speed of light in the medium) or if the current goes through all possible measures in the range from off to on.
Thank you very much!! :) I understand discontinuities and continuities, Just wasnt sure how to approach them, your explanations made it much clearer!!!