1. It is one of the most important tasks of analysis to compute the number
iV(/, fi, yo) of solutions x E fi of the equation
f(x) = 2/o, (1)
where fl is a subset of some domain space E and where yo is a point of a range
If 0 is the open interval (a, b) on an x-axis, if / is a polynomial in x, and
if yo is a point on a y-axis, then the above problem is completely solved by
a classical theorem of Sturm (1835) which gives a procedure to calculate the
However, this number may not be continuous in dependence on yo or /. If,
e.g., fi is the interval (-1, +1) on the real axis R and / is the map fi — R given
by f(x) = y =
then for any positive e 1
2 if yo = e,
1 ifyo = 0,
0 if yo = -£.
Thus our number N is not continuous in yo at yo = 0. It is just as easy to see
that iV(/, fi,yo) may not change continuously if / is changed continuously. For
instance if fi is as above and f(x) =
+ a with a real, then iV(/,fi,0) is not
continuous at a = 0.
2. The basic difference between the root count iV(/, fi,yo) and the "degree"
d(/fyyo) is that the latter has the above-mentioned continuity properties.
There are two good reasons for insisting on these: (i) without them a small
error in the computation of yo or / may lead to a wrong root count; (ii) if
d(/, fi,yo) is continuous in a certain class C of mappings /, then this number,
being integer-valued, is constant for all / E C. It may happen that C contains an
/o for which the degree is particularly easy to compute. A case often occurring
in the applications is that C contains the identity map /o = J.
3. We return to the example y = f(x) = x2 of §1 to indicate how a root count
d(fi ^ 2/o) niay be obtained which is continuous in yo at yo = 0. For yo 0 we