 A pioneer in the matter of clarifying the concept of continuous function was the Bohemian priest, philosopher and mathematician Bernard Bolzano (1781–1848). In his Rein analytischer Beweis of 1817 he defines a (real-valued) function \(f\) to be continuous at a point \(x\) if the difference \(f(x + \omega) - f(x)\) can be made smaller than any preselected quantity once we are permitted to take \(w\) as small as we please. This is essentially the same as the definition of continuity in terms of the limit concept given a little later by Cauchy. Bolzano also formulated a definition of the derivative of a function free of the notion of infinitesimal (Bolzano 1851 [1950]). Bolzano repudiated Euler’s treatment of differentials as formal zeros in expressions such as \(\Dy/\Dx\), suggesting instead that in determining the derivative of a function, increments \(\Delta x,\) \(\Delta y,\) …, be finally set to zero. For Bolzano differentials have the status of ideal elements, purely formal entities such as points and lines at infinity in projective geometry, or (as Bolzano himself mentions) imaginary numbers, whose use will never lead to false assertions concerning real quantities.