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*If ''y'' is rational, then ''f''(''y'') = 1. To show the function is not continuous at ''y'', we need to find an ε such that no matter how small we choose δ, there will be points ''z'' within δ of ''y'' such that ''f''(''z'') is not within ε of ''f''(''y'') = 1. In fact, 1/2 is such an ε. Because the [[irrational number]]s are [[dense set|dense]] in the reals, no matter what δ we choose we can always find an irrational ''z'' within δ of ''y'', and ''f''(''z'') = 0 is at least 1/2 away from 1.
*If ''y'' is irrational, then ''f''(''y'') = 0. Again, we can take ε = 1/2, and this time, because the rational numbers are dense in the reals, we can pick ''z'' to be a rational number as close to ''y'' as is required. Again, ''f''(''z'') = 1 is more than 1/2 away from ''f''(''y'') = 0.
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The ''Dirichlet function'' can be constructed as the double pointwise limit of a sequence of continuous functions, as follows:
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