I am reading Mathematical Statistics for Economics and Business by Mittelhammer he stated on page 59 that this function $$ f(x)=(0.3)^x(0.7)^{1-x}I_{[0,1]}(x)$$ could serve as discrete probability density function .
My qustion is , how this function could be a discrete probability density function when $$f(x)\neq 0 \forall x \neq {0, 1} ?$$
The factor, $I[\{0,1\}](x)$ , is an indicator function; a piecewise function having the value of $1$ when $x\in\{0,1\}$ and $0$ elsewhere. This is what ensures that $f(x)$ is zero elsewhere.
$$I[\{0,1\}](x) = \begin{cases}1 & : x= 0\vee x=1 \\ 0 & : \textsf{elsewhere}\end{cases}$$