If f(x) = 2 x<0
1 x=0
x^2 x>0
the lim x-0- f(x) = 2
the lim x-0+ f(x) = 0
right?
So lim x-0 does not exist as they are not equal?
But as f(0) is defined as 1 does this effect the limits? or do we ignore this?
For a limit to exist at a point, the left and right hand limits need to be equal. So you are correct that the limit does not exist.
For a function to be continuous at a point, the function must be defined at a point, the limit must exist (so the left and right hand limits must be equal), and the limit value needs to be the same as the function value. So the function is also discontinuous.