Question

Explain continuity in limits: how does the thm below work?

Answer 1

specific example would be good

Answer 2

definition of continuity is
\[f\text { is continuous at } c \text{ if } \lim_{x\rightarrow c} f(x)=f(c)\]

Answer 3

which means that f exists at c, and that the two sided limit exists, and they are equal

Answer 4

Yeah, I know that the thm goes like that but doesn't it just mean that f(x) approaches f(c) but doesn't necessarily meet it?

Answer 5

How do you know that f(x) doesn't have a hole?

Answer 6

more detailed would be f is continuous at c if \[\lim_{x \rightarrow c+}f(x)=a, \lim_{x \rightarrow c-}f(x)=b\] and \[a=b\]

Answer 7

if f had a whole then when you approach the function from the left would give you a different value than the one you'd obtain by approaching the function from the right.

Answer 8

because if f has a "hole" then the function doesn't exist there

Answer 9

in other words if
\[(c,f(c))\] is not on the graph, then
\[f(c)\] does not exist. part of the definition of continuity is that the function is defined at that point

Answer 10

well, what if f(x) was a removable discontinuity? i mean what if (c, f(c)) existed but not where it should for f(x) to be continuous?

Answer 11

i see the confusion. there are three parts of continuity.
1) the function exists at the number (so there cannot be a hole there)
2) the limit exists at the number
3) the value of the function is the same as the limit

Answer 12

example of limit existing, but function not existing at a point c, so not continuous because
\[f(c)\] does not exist

Answer 13

example of function existing, limit existing, but not continuous because they are not equal

Answer 14

right, so how does \[\lim_{x \rightarrow c} f(x) = f(c)\] explain that f(c) is where it should be?

Answer 15

example of function existing, but not continuous because limit does not exist

Answer 16

"where it should be" is a good english way to think about it, but the math is what is written above.

Answer 17

I mean, say \[\lim_{x \rightarrow 4} = 5\] that doesn't mean that f(x) necessarily is continuous

Answer 18

oops forgot the f(x)

Answer 19

if your function is the constant function
\[f(x)=5\] then it is certainly continuous because it 5 for all values of x

Answer 20

\[\lim_{x\rightarrow 4}f(x)=5=f(4)\] and your function is continuous at 4.

Answer 21

oh i mis understood.

Answer 22

ok, I'm gonna think about this all over again. Maybe I just need to rethink. Thanks so far though ^^

Answer 23

look at the examples i drew above. each one violates some condition of continuity. if the function is continuous at a number c, it means that (in essence) when you go to draw it you do not have to lift you pencil when you get to c

Answer 24

no hole, no jump, no going to infinity, and the limit from the left is equal the limit from the right

Answer 25

Another way to see it is that small variations in the x axis mean small variations in the y axis too, if you have that the function is continuous.

Answer 26

ohhh. I see the greatest factor in my confusion: the \[, x \neq 4\] that I missed with all of the discontinuous f's. :P

Answer 27

Well thanks guys for helping me get rid of this self-induced confusion :P