I don't use {array} (normally) but {aligned} which like {align} (for separate equations left-aligns odd tabs (&) and right-aligns even tabs. I would use

`$$`

f(x) = \left\{\begin{aligned}

&1 &&: |x| \le 1\\

&0 &&: |x| > 1

\end{aligned}

\right.$$

producing

$$

f(x) = \left\{\begin{aligned}

&1 &&: |x| \le 1\\

&0 &&: |x| > 1

\end{aligned}

\right.$$

With array I would use

`$$f(x) = \left\{\begin{array}{ll}`

1 & : |x| \le 1\\

0 & : |x| > 1

\end{array}

\right.$$

producing

$$f(x) = \left\{\begin{array}{ll}

1 & : |x| \le 1\\

0 & : |x| > 1

\end{array}

\right.$$

Note that in {array} like in {tabular} l, r, c indicate type of alignment of the corresponding column (and one can put | between them to produce a vertical line)