by Intermediate Value theorem
, there is a point 'x' such that g(x)=0 that is f(x) =x.
Since there is no intermediate value theorem
for complex valued functions, the proof does not carry over to the case B [subset] C, though B may be regarded as two dimensional in this case.
A good illustration here is the Intermediate Value Theorem, for instance the special case (known as Bolzano's theorem) which states that any continuous function f defined on the interval [0, 1] satisfying f(0) = -1 and f(1) = +1 has a zero, i.e.
Here, as in the case of the Intermediate Value Theorem, there are constructive substitutes, but they are essentially weaker, in this case significantly so.
Indeed, u has the largest zero [??] [member of] (a, b) and it follows from the Lagrange intermediate value theorem
that there is [bar.a] [member of] ([??], b) such that u'([bar.a]) = 0.
Thus by intermediate value theorem
, [r.sub.0] exists in the interval 0 < [r.sub.0] < [r.sub.1]
By the intermediate value theorem
, there exists a pair u, v such that D [u, v] = 0 and hence F is not interpolating at these points.
Bernard Bolzano proved the intermediate value theorem. This was early in the nineteenth century, and commentators since typically say two things: first, that Bolzano's work was initially unappreciated and only later brought to light or rediscovered by others such as Cauchy and Weierstrass; second, that thanks to Bolzano and the others, we now have a rigorous proof of the theorem, whereas before we only had a good hunch based on a geometrical diagram.
Most calculus and analysis texts contain a proof of the intermediate value theorem, and often they have a few casual comments about its significance.
When this attitude is brought to bear on the intermediate value theorem, it is perfectly natural to conclude that, until Bolzano, we couldn't really be sure the theorem is true.