how to determine the intervals on which the function is continuous in a piece-wise function

Question

anonymous · Answer

You must plot the pieces and see were it is continuous. Be careful of holes.
I.E. (-inf, 8)U(8, inf) is not continuous, do you see/know why?

anonymous · Answer

(-10, 0]U(0, 10) is not continuous either... left side is including 0, right side never touches 0

debbieg · Answer

I don't think you necessarily have to graph it. If the equation defining each piece is continuous, then you only need to check continuity at the endpoints of each piece.

E.g., if you have:

f(x)=x+1  for x>0
       =x^2+1 for x<=0

then each of those functions are continuous, so you only need to check that the limit as x->0 exists and is =f(0). If so,t he function is continuous.

debbieg · Answer

@ehuman I'm not sure what you mean.... you are talking about intervals of the domain, I guess? If we are looking at a piecewise function, then certainly, if there is a "hole" in the domain, an x value for which the function is not defined, then agreed - that will be a point of discontinuity.

But the harder part for these kinds of problems is usually to look at the points in the domain where the rule for the function changes. The limits from the left and right must be equal, and must equal the value of the function at that point.