Nowhere-Differentiable, Desert Mug

Calculus instructors often use the phrase ``continuous function'' (loosely, one whose graph has no jumps) while drawing a differentiable function (one whose graph has a tangent line at each point). It can therefore come as a conceptual shock to learn in real analysis that a continuous function may be differentiable nowhere, and in fact that a ``randomly chosen'' continuous function is nowhere-differentiable.

A sequence of smooth (infinitely-differentiable) functions can converge to a nowhere-differentiable function. In the image, a sequence of trigonometric polynomials converges ``uniformly'' to a Weierstrass nowhere-differentiable function. Successive terms look like ranges of ever-more-distant hills.

Printed on a sturdy cylindrical 12 oz. (350 mL) white ceramic mug with handle. Microwave and dishwasher safe.