  Weyl also agreed with Brouwer that all functions everywhere defined  on a continuum are continuous, but here certain subtle differences of viewpoint emerge. Weyl contends that what mathematicians had taken to be discontinuous functions actually consist of several continuous  functions defined on separated continua. In Weyl’s view, for example,  the “discontinuous” function defined by  \(f(x) = 0\) for \(x \lt 0\) and  \(f(x) = 1\) for \(x \ge 0\) in fact consists of  the two functions with constant values and 1 respectively  defined on the separated continua \(\{x: x \lt 0\}\) and \(\{x: x \ge 0\}\). (The union of these two continua  fails to be the whole of the real continuum because of the failure of the law of excluded middle: it is not the case that, for be any real  number \(x\), either \(x \lt 0\) or \(x \ge 0\).)  Brouwer, on the other hand, had not dismissed the possibility that  discontinuous functions could be defined on proper parts of a  continuum, and still seems to have been searching for an appropriate  way of formulating this   idea.[23]  In particular, at that time Brouwer would probably have been inclined to regard the above function \(f\) as a genuinely discontinuous  function defined on a proper part of the real continuum. For Weyl, it seems to have been a self-evident fact that all functions defined on  a continuum are continuous, but this is because Weyl confines  attention to functions which turn out to be continuous by definition. Brouwer’s concept of function is less restrictive than Weyl’s and it  is by no means immediately evident that such functions must always be continuous. 