Zero variance Random variables

I am a probability theory beginner. The expression for the variance of a random variable $x$ (of a random process is

$$\sigma^2 = E(x^2) – (\mu_{x})^2$$

If $E(x^2) = (\mu_{x})^2$, then $\sigma^2 = 0$. Can this happen ? Can a random variable have a density function whose variance (the second central moment alone) is $0$ (other than the dirac delta function).

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