A function is continuous at a point x=a if the function evaluated at a, i.e. f(a), equals both the limit of f(x) as x approaches a from the left and as x approaches a from the right
In math symbols, the function is continuous at a if
I have a couple of questions regarding continuity. One is just a general question and the second is a textbook question.
1) Can anyone give me a quick explanation in regards to the concept of continuity? We transitioned from limits (which I picked up really well) into continuity and I'm having a tough time distinguishing between the two.
2) The directions state: Find all values of x = a where the function is continuous. For each value of x, give the limit of the function as x approaches a.
The Problem is: f(x) = 5 + x/x(x-2)
I'm ending up with 5 + x/-2, but I'm completely lost as to the next step in the problem.
Any help would be greatly appreciated. Thanks!
A function is continuous at a point x=a if the function evaluated at a, i.e. f(a), equals both the limit of f(x) as x approaches a from the left and as x approaches a from the right
In math symbols, the function is continuous at a if
continuity and limits are closely linked.
as the previous poster noted, f is continuous at a if lim x-->a f(x) = f(a).
this means two things:
1) the limit lim x-->a f(x) has to exist (the right-hand and left-hand limits must both exist, AND be equal).
2) f(a) has to exist.
for example, the function f(x) = x/x, has a perfectly good limit at x = 0: lim x-->0 x/x = 1.
however, the function is NOT continuous there, as f(0) is undefined (0/0 has no well-defined meaning).
now what it means for lim x-->a f(x) = f(a) is this: as we get close to a, f doesn't do anything unexpected, such as have a hole, or suddenly switch values, or become infinitely large in size.
put another way: if x is near a, then f(x) is near f(a) (the epsilons and deltas are for making precise this notion of "near").
your function (which i hope is) f(x) = -5 + x/(x(x-2)), has two discontinuities: one at x = 0 (it has a "hole" there, because it is undefined, even though the limit lim x-->0 f(x) exists),
and another at x = 2, where f becomes unbounded (this is another way of saying it becomes "infinitely big" in size).