Are 1/x and 1/x^2 continuous functions?

continuous-functions

#1

Are 1/x and 1/x^2 continuous functions?

Answer:

No, because when x=0 there is a whole on the graph, and for a function to be continuous there can not be a whole or an asymptote.

No, there are 3 steps to determine continuity

  1. The function at a certain value must be defined
    ->1/x and 1/x^2 fail that test at x=0

  2. The limit must approach the same value on both sides
    ->1/x approaches -inf. and inf at x=0
    ->1/x^2 technically approach infinity from both sides at x=0, but infinity isn’t a number, so 1/x^2 fails that test as well

  3. The limit must be equal to the function value
    ->You can already see how this is automatically untrue for both functions since they’re both undefined at x=0.

If a function fails ANY of these tests, the function isn’t continuous. (And 1/x and 1/x^2 fail all these steps)

TL;DR: No, because there’s an asymptote at x=0