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The discontinuity of probability mass functions reflects the fact that the cumulative distribution function of a discrete random variable is also discontinuous. Where it is differentiable (i.e. where ''x'' ∈ '''R'''\''S'') the derivative is zero, just as the probability mass function is zero at all such points.
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Suppose that ''X'' is the outcome of a single coin toss, assigning 0 to tails and 1 to heads. The probability that ''X'' = ''x'' is 0.5 on the state space {0, 1} (this is a [[Bernoulli distribution|Bernoulli random variable]]), and hence the probability mass function is
:<math>f_X(x) = \begin{cases}\frac{1}{2}, &x \in \{0, 1\},\\0, &x \in \mathbb{R}\backslash\{0, 1\}.\end{cases}</math>
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