The more interesting statement is: > There exist smooth functions of a single real variable that have derivatives at the origin that are all zero, yet are nonz

The more interesting statement is: > There exist smooth functions of a single real variable that have derivatives at the origin that are all zero, yet are nonzero https://en.wikipedia.org/wiki/Non-analytic_smooth_function https://en.wikipedia.org/wiki/Non-analytic_smooth_function