pathological curve

A pathological curve is a curve often specifically devised to show the falseness of certain intuitive concepts. In particular, the image of continuity as a smooth curve in our mind's eye severely misrepresents the situation and is the result of a bias stemming from an overexposure to the much smaller class of differentiable functions. A chief lesson of pathological curves is that continuity is a weaker notion than differentiability. Many pathological curves are fractals, such as Cantor dust, including space-filling curves, such as the Peano curve. The earliest known example is the graph of Weierstrass' non-differentiable function.