A Singular Surface In Three Dimensions
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This animation depicts a surface which has a peculiar singularity.
(See notes below.)
The surface shown here is given by the equation
z = 5 x2 y ⁄ (x4 + y2).
The singularity at the origin has a very interesting
nature: The limit of z as (x, y) → (0, 0)
along lines of the form y = m x, where we have
z = 5 m x ⁄ (x2 + m2), is always zero. Nevertheless, the limit
of z as (x, y) → (0, 0) does not exist. To see this consider what happens to z
as (x, y) → (0, 0) along parabolae of the form y = m x2.
Along such parabolae, we have z = 5 m ⁄ (1 + m2).
If we put z = 0 at the origin, we obtain an
example of a function which is discontinuous at the origin but for which
the directional derivatives at the origin exist in every direction.