A posterior formula that reproduces both point masses of a two-point prior

Suppose given $\theta$, the random variable X follows a normal distribution with mean $\theta$ and variance 1. Let the prior density of $\theta$ be Bernoulli with parameter $1/2$ (i.e. $\theta\in \{0,1\}$ each with prior probability $1/2$). Then the posterior distribution of $\theta$ is: