is a straight line with a wave laid over it. It is a standard example in calculus of a function whose derivative vanishes at certain points and yet which has no extremum anywhere.
Both the domain and the range are all of the real numbers. Since , the function is odd and the graph has rotational symmetry about the origin.
The derivative is as follows.
Because we always have , so the function increases monotonically. It vanishes where , that is at , but those are isolated points and the increase never actually stops.
This function shows plainly that a vanishing derivative does not imply an extremum: the values keep rising through , and only the tangent line goes momentarily flat.
The second derivative is , which changes sign at every . The inflection points are therefore at , where . All of them, the points , lie on the line .
At odd multiples of the tangent is horizontal, while at even multiples it has slope , so flat and sloping inflection points alternate.
| Slope of the tangent | ||
|---|---|---|
For every integer the point at is .
Since , the graph oscillates about the line with amplitude , crossing it at every .
The difference does not tend to , however, so is not an asymptote but merely the center line of the oscillation. The contrast with a genuine slant asymptote, as in where the gap does vanish, is easy to see here.
Being continuous, strictly increasing and onto the whole real line, the function has an inverse defined for every real number. At the points corresponding to , where , the graph of that inverse has a vertical tangent.
Changing the sign gives , the same curve translated. That form appears as the -coordinate of the cycloid1.
Kepler's equation approaches exactly this shape for orbits whose eccentricity is close to 2. Finding a body's position from the time means solving numerically a function that is increasing yet has points of zero slope, and that is precisely where the computation becomes delicate.