We are now in a position to consider the extension of Bolzano's theorem in soft setting.

In this paper we have dealt with continuity and differentiability of functions of soft real sets and extended some celebrated theorems, like Bolzano's theorem, fixed point theorem, intermediate value property, and Rolle's theorem, in soft settings.

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.

One may feel that the visual route to the theorem can be defended against this argument in the following way: the counter-examples to the link assumption that we have described are curves of discontinuous functions; since Bolzano's theorem is about continuous functions only, we can ignore these curves.

So Bolzano's Theorem cannot be discovered by visualizing in the manner described above.(9)

Consider the following route to the Intermediate Value Theorem (IVT) from Bolzano's Theorem. IVT is this: Let g be a continuous function over [a, b] such that g(a) [not equal to] g(b).

Assuming that mere translation preserves continuity, the newly positioned curve is the curve of a function which satisfies the conditions of Bolzano's Theorem, and so its curve meets the x-axis at some point between a and b.

For one thing, if they are totally unrelated concepts then it would make no sense to even illustrate

Bolzano's theorems with the diagrams, nor would it make any sense to apply Bolzano's result to situations in geometry or mechanics, as is commonly done.