## anonymous 4 years ago Explain continuity in limits: how does the thm below work?

1. anonymous

specific example would be good

2. anonymous

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

3. anonymous

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

4. anonymous

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?

5. anonymous

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

6. anonymous

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 $a=b$

7. anonymous

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.

8. anonymous

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

9. anonymous

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

10. anonymous

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?

11. anonymous

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

12. anonymous

|dw:1327164145121:dw| example of limit existing, but function not existing at a point c, so not continuous because $f(c)$ does not exist

13. anonymous

|dw:1327164195706:dw| example of function existing, limit existing, but not continuous because they are not equal

14. anonymous

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

15. anonymous

|dw:1327164250426:dw| example of function existing, but not continuous because limit does not exist

16. anonymous

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

17. anonymous

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

18. anonymous

oops forgot the f(x)

19. anonymous

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

20. anonymous

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

21. anonymous

oh i mis understood.

22. anonymous

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

23. anonymous

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

24. anonymous

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

25. anonymous

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.

26. anonymous

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

27. anonymous

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