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This contrasts with classical analysis, which (in this context) simply means analysis done according to the (ordinary) principles of classical mathematics.
Generally speaking, constructive analysis can reproduce theorems of classical analysis, but only in application to separable spaces; also, some theorems may need to be approached by approximations. Furthermore, many classical theorems can be stated in ways that are logically equivalent according to classical logic, but not all of these forms will be valid in constructive analysis, which uses intuitionistic logic.
For a simple example, consider the intermediate value theorem (IVT). In classical analysis, IVT says that, given any continuous function f from a closed interval [a,b] to the real line R, if f(a) is negative while f(b) is positive, then there exists a real numberIn mathematics, the real numbers are intuitively defined as numbers that are in one-to-one correspondence with the points on an infinite line—the number line. The term "real number" is a retronym coined in response to " imaginary number". Real numbers may c in the interval such that f(c) is exactly zero. In constructive analysis, this does not hold, because the constructive interpretation of existential quantificationLogic In predicate logic, existential quantification is an attempt to formalize the notion that something (a logical predicate) is true for something or at least one relevant thing. The resulting statement is an existentially quantified statement, and we ("there exists") requires one to be able to construct the real number c (in the sense that it can be approximated to any desired precision by a rational numberIn mathematics, a rational number (or informally fraction is a ratio of two integers, usually written as the vulgar fraction a ''b where b is not zero. The set of all rational numbers is denoted by Q or in blackboard bold. Using the set-builder notation i). But if f hovers near zero during a stretch along its domain, then this cannot necessarily be done.
However, constructive analysis provides several alternative formulations of IVT, all of which are equivalent to the usual form in classical analysis, but not in constructive analysis. For example, under the same conditions on f as in the classical theorem, given any natural numberNatural number can mean either a positive integer ( 1, 2, 3, 4,. or a non-negative integer ( 0, 1, 2, 3, 4,. Natural numbers have two main purposes: they can be used for counting ("there are 3 apples on the table"), or they can be used for ordering ("this n (no matter how large), there exists (that is, we can construct) a real number cn in the interval such that the absolute valueIn mathematics, the absolute value (or modulus of a number is that number without a negative sign. So, for example, 3 is the absolute value of both 3 and −3. Definition It can be defined as follows: For any real number a the absolute value of a deno of f(cn) is less than 1/n. That is, we can get as close to zero as we like, even if we can't construct a c that gives us exactly zero.
Alternatively, we can keep the same conclusion as in the classical IVT -- a single c such that f(c) is exactly zero -- while strengthening the conditions on f. We require that f be locally non-zero, meaning that given any point x in the interval [a,b] and any natural number m, there exists (we can construct) a real number y in the interval such that |y - x| < 1/m and |f(y)| > 0. In this case, the desired number c can be constructed. This is a complicated condition, but there are several other conditions which imply it and which are commonly met; for example, every analytic functionIn mathematics, an analytic function is one that is locally given by a convergent power series. Complex analysis teaches us that if a function f of one complex variable is differentiable in some open disk D centered at a point c in the complex field, then is locally non-zero (assuming that it already satisfies f(a) < 0 and f(b) > 0).
For another way to view this example, notice that according to classical logic, if the locally non-zero condition fails, then it must fail at some specific point x; and then f(x) will equal 0, so that IVT is valid automatically. Thus in classical analysis, which uses classical logic, in order to prove the full IVT, it is sufficient to prove the constructive version. From this perspective, the full IVT fails in constructive analysis simply because constructive analysis does not accept classical logic. Conversely, one may argue that the true meaning of IVT, even in classical mathematics, is the constructive version involving the locally non-zero condition, with the full IVT following by "pure logic" afterwards. Some logicians, while accepting that classical mathematics is correct, still believe that the constructive approach gives a better insight into the true meaning of theorems, in much this way.