Consider the step function $$ \Delta(x;\lambda,\mu)\equiv \sum_{j=1}^J \lambda_j\times 1\{\mu_j\leq x\} $$ where
$\lambda_j\geq 0$ $\forall j$; $\sum_{j=1}^J \lambda_j=1$
$\mu_j\in \mathbb{R}$ $\forall j$; $\mu_1<...<\mu_J$
$1\{\cdot\}$ is an indicator function taking value $1$ if the condition inside is satisfied and zero otherwise
$\lambda\equiv (\lambda_1,...,\lambda_J)$
$\mu\equiv (\mu_1,...,\mu_J)$
Take a random variable $Y$ with CDF given by $\Delta(\cdot; \lambda,\mu)$. Is it correct to say that $Y$ should be necessarily a discrete random variables with support $\{\mu_1,...,\mu_J\}$ and with probabilities masses equal to $\{\lambda_1,...,\lambda_J\}$?