In the late 19th century, Karl Weierstrass invented a fractal-like function that was decried as nothing less than a “deplorable evil.” In time, it would transform the foundations of mathematics.

If \(f\) is differentiable on \([-1,1]\) then \(f\) is continuous at \(x=0\text{.}\) If \(f'(x)\lt 0\) and \(f"(x)>0\) for all \(x\) then \(f\) is concave down ...