Self-adjoint operator proportional to identity if and only if the support of the spectral measure is a singleton

Question

Let A be a self-adjoint operator on a Hilbert space $\mathcal{H}$, and let $P_A$ be the projection valued measure (spectral measure) obtained from the spectral theorem, such that

$A=\int \lambda\ dP_A(\lambda)$

How do I show that

$A=\lambda I$ $\ \ \ $ $\Leftrightarrow$ $\ \ \ $ $\text{supp}P_A=\{\lambda\}$