 We observe that the postulates of smooth infinitesimal analysis are incompatible with the law of excluded middle of classical logic. This incompatibility can be demonstrated in two ways, one informal and the other rigorous. First the informal argument. Consider the function \(f\) defined for real numbers \(x\) by \(f(x) = 1\) if \(x = 0\) and \(f(x) = 0\) whenever \(x \ne 0\). If the law of excluded middle held, each real number would then be either equal or unequal to 0, so that the function \(f\) would be defined on the whole of \(\bR\). But, considered as a function with domain \(\bR,\) \(f\) is clearly discontinuous. Since, as we know, in smooth infinitesimal analysis every function on \(\bR\) is continuous, \(f\) cannot have domain \(\bR\)  there.[50]  So the law of excluded middle fails in smooth infinitesimal analysis. To put it succinctly, universal continuity implies the failure of the law of excluded middle.