Question

How do I know if a function is continuos? This question got me stuck:
f(x)={x^2-3, if x<-1
{sqrt(3-x), if x>=1
( Also I have to include limit notations)

Answer 1

are you sure it's "x<-1" and it's not "x<1"

Answer 2

Yes I'm sure

Answer 3

is " x>=1" supposed to be "x>=-1" ?

Answer 4

Oh yeah i accidently wrote it in positive.
i meant for it to be negative also

Answer 5

ok that makes more sense

Answer 6

if the function is continuous, then the two pieces must connect

Answer 7

this only happens when the function value at the endpoint is the same

so plug in x = -1 into x^2-3

and plug x = -1 into sqrt(3-x)

if you get the same result, then the function is continuous at x = -1

Answer 8

Ohhh. I see what you mean now. Thank you for that clarification :)
and what about using limit notations?

Answer 9

well the idea is that if the left hand and right hand limits at the endpoint equal the same value, then the limit exists at this point

so if you can prove that the function exists at x = -1, the two pieces connect, and the two limits are equal, then you will have shown that the function is continuous here

Answer 10

here's a visual example, say we had this piecewise function