Question

How do you explain why `f(x) = x^2` if `x>=1` and 2x if x<1 is continuous at x = 1?

Answer 1

It is not continuous at `1`, because the limit at `1` does not exist.

Explanation:

`f(x) = {(x^2,"if",x >=1),(2x,"if",x < 1) :}`

`f` is continuous at `1` if and only if

`lim_(xrarr1)f(x) = f(1)`. `" "` (existence of both is implied)

We see from the definition of `f` above that

`f(1) = (1)^2 = 1`.

In finding the limit, we'll need to consider the left and right limits separately because the rule changes at `x=1`.

The limit from the right is

`lim_(xrarr1^+)f(x) = (1)^2 = 1`.

The limit from the left is

`lim_(xrarr1^-)f(x) = 2(1) = 2`.

Because the right and left limits are not the same, the limit does not exist.

Therefore the function is not continuous at `1`.