Fraktal

 Generate a computable number.

  Problem: Let be a function defined on the real line such that for every . Prove for all , i.e. is constant.   Show Solution Solution: Take any . Next, we obtain for all . So, for , . Then, take the limit as approaches , we get   Hence, the derivative is zero 0 everywhere. Therefore, must be a constant function.

Let I be a closed interval and f : I → ℝ be a differentiable function. Then the derivative f' of f has the intermediate value property : if a , b ∈ I with a < b , then for every y ∈ ℝ such that  f' ( a ) < y < f' ( b ) or f' ( b ) < y < f' ( a ), there exists x ∈ [ a , b ] such that f' ( x ) = y .

Let A be a subset of the set of all real numbers ℝ. A real number c is called the infinum of A ( c = inf A ) if satisfies:

 A sequence of real function ( f n ) is said uniformly convergent on a set A⊆ℝ to a limit function f :A→ℝ if and only if for every 𝜺 > 0 there exists N ∈ ℕ such that for every natural number n ≥ N and for every x ∈ A,