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

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

Mathematics
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specific example would be good
definition of continuity is \[f\text { is continuous at } c \text{ if } \lim_{x\rightarrow c} f(x)=f(c)\]
which means that f exists at c, and that the two sided limit exists, and they are equal

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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?
How do you know that f(x) doesn't have a hole?
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 \[a=b\]
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.
because if f has a "hole" then the function doesn't exist there
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
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?
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
|dw:1327164145121:dw| example of limit existing, but function not existing at a point c, so not continuous because \[f(c)\] does not exist
|dw:1327164195706:dw| example of function existing, limit existing, but not continuous because they are not equal
right, so how does \[\lim_{x \rightarrow c} f(x) = f(c)\] explain that f(c) is where it should be?
|dw:1327164250426:dw| example of function existing, but not continuous because limit does not exist
"where it should be" is a good english way to think about it, but the math is what is written above.
I mean, say \[\lim_{x \rightarrow 4} = 5\] that doesn't mean that f(x) necessarily is continuous
oops forgot the f(x)
if your function is the constant function \[f(x)=5\] then it is certainly continuous because it 5 for all values of x
\[\lim_{x\rightarrow 4}f(x)=5=f(4)\] and your function is continuous at 4.
oh i mis understood.
ok, I'm gonna think about this all over again. Maybe I just need to rethink. Thanks so far though ^^
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
no hole, no jump, no going to infinity, and the limit from the left is equal the limit from the right
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.
ohhh. I see the greatest factor in my confusion: the \[, x \neq 4\] that I missed with all of the discontinuous f's. :P
Well thanks guys for helping me get rid of this self-induced confusion :P

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