Question

Find the numbers, if any, where the function is discontinuous. f(x) = [(t+2)^1/2]/[(t+1)^1/2]

Isn't it true that (-inf, -1] is the set of numbers that x is discontinuous at? My book says that there are no discontinuous numbers as the answer, so I am extremely confused, because radicands of a square root cannot be negative and denominators cannot be zero if a function is to be defined at all points (and thus continuous)?

Answer 1

are you sure it's not f(t) instead of f(x)?

Answer 2

Oh yeah, sorry, f(t) my bad.

Answer 3

f(t) isn't even defined in (-inf, -1] so you can't say anything about this interval.
What matters is where the function is defined, and that is on (-1,inf). On this interval, the function is continuous

Answer 4

Oh I see, so when it asks about continuity, all that matters is that it is at least continuous on some interval that goes to either infinity or -infinity? As in, it doesn't matter where the end-point is?

Answer 5

informally speaking, a continuous has no jump where it's defined

Answer 6

So for something like f(x) = [sin(x)]/[x], it would be undefined at +/-pi*n (where n is an integer) because this would be a removeable discontinuity in the middle, breaking up the two continuous segments?

Answer 7

sin(x)/x *is* continuous. Continuity only deals with where the function is defined.

however if you have a function like:
f(x) = sin(x)/x , when x ≠0 and 10 when x = 0, then now the function is not continuous because f(0) = 10, but lim x->0 f(x) ≠ 10 (in fact, the limit is 1)

Answer 8

All right, I think I understand it better now. Thanks for your help.