Question

Determine the intervals on which the function is continuous.

f(x) (5-x, x<=2) and (2x-3, x>2)

Answer 1

would it just be all real numbers? or all besides x=2 since the graphs don't touch at 2?

Answer 2

\(5-x\) and \(2x-3\) are continuous for their respective pieces, but you have to check if \(\displaystyle\lim_{x\to2}f(x)\) exists. To do that, you have to check the one-sided limits, since the function takes on \(5-x\) for values less than 2 and \(2x-3\) takes on values greater than 2.

Answer 3

Basically, if \(\displaystyle\lim_{x\to2^-}f(x)=\lim_{x\to2^+}f(x)\), then \(f(x)\) is continuous. If they're not equal, \(f(x)\) is not continuous at 2.

Answer 4

I see. So in this case the lmiits of the functions would be 3, and 1 respectively. Since they're not equal, not continuous at 2? Just checking.

Answer 5

Yes, that's right.

Answer 6

Thanks!

Answer 7

You're welcome!