Q: Notation for Differentiable Functions

[rudders93]

Hi,

I was wondering, how can i express the following in notation (function notation i think it is? The one where we {(x,y) $$\in$$ R³ : x + y = 0}

Q: Set of real polynomials of any degree

Q: Set of all differentiable functions (which I guess just means continuous functions, but nevertheless not sure how to express this properly?)

Thanks!

 

[Simon_Tyler]

Hi rudders93,
I hope the following is what you are looking for (and free from errors)

The space of polynomials in a formal variable x over the field F is denoted
F[x] = { a₀ + a₁x +...+ a_nxⁿ | a_i∈F, n < ∞ }
See http://en.wikipedia.org/wiki/Examples_of_vector_spaces#Polynomial_vector_spaces" on Wikipedia for more info.

Continuous does not mean the same as differentiable.
The set of real functions for which the n^th derivative exists over the entire domain Ω⊆R is denoted
Cⁿ(Ω) = { f:Ω→R | f'∈C^n-1(Ω) }
This is a recursive definition terminating with continuous functions
C⁰(Ω) = { f:Ω→R | lim_x→cf(x)=f(c) ∀ c∈Ω }
For more info see Wikipedia:
http://en.wikipedia.org/wiki/Smooth_function#Differentiability_classes"
http://en.wikipedia.org/wiki/Continuous_function"

 

[rudders93]

Thanks!