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\begin{figure}\begin{center}\BoxedEPSF{Cusp.epsf scaled 700}\end{center}\end{figure}

A cusp is a point on a continuous curve where the tangent vector reverses sign as the curve is traversed. A function $f(x)$ has a cusp (also called a Spinode) at a point $x_0$ if $f(x)$ is Continuous at $x_0$ and $\lim_{x\to x_0} f'(x) = \infty$ from one side while $\lim_{x\to x_0} f'(x) = -\infty$ from the other side, so the curve is Continuous but the Derivative is not. A cusp is a type of Double Point. The above plot shows the curve $x^3-y^2=0$, which has a cusp at the Origin.

See also Double Cusp, Double Point, Ordinary Double Point, Ramphoid Cusp, Salient Point


Walker, R. J. Algebraic Curves. New York: Springer-Verlag, pp. 57-58, 1978.

© 1996-9 Eric W. Weisstein