Nowhere-Differentiable, Sun Greeting Card

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 such as trigonometric polynomials can converge to a nowhere-differentiable function, such as a ``Weierstrass function.'' Successive terms of the sequence look like ranges of ever-more-distant hills.

Printed on high-quality 330gsm Fedrigoni card stock.