  We discuss some surprising phenomena from basic calculus related to
oscillating functions and to the theorem on the differentiability of inverse
functions. Among other things, we see that a continuously differentiable
function with a strict minimum doesn't have to be decreasing to the left nor
increasing to the right of the minimum, we present a function whose derivative
is discontinuous at one point and has a strict minimum at this point (i.e. it
oscillates only in one direction), we compare several definitions of inflection
point, and we discuss a general version of the theorem on the derivative of
inverse functions where continuity of the inverse function is assumed merely at
one point.
