Defining of function in equation

[matematikuvol]

$$\varphi(x)=f(x)+\int^{b}_{a}K(x,y)\varphi(y)dt$$

$$f:[a.b]→ℝ$$
$$K(x,y)→[a,b]\times [a,b]→ℝ$$

Is $$[a,b]\times [a,b]$$ Deckart product? Is that the way to construct $$ℝ^2$$ space?

If I say $$f\in C([a,b])$$, $$K\in C([a,b]\times [a,b])$$ that means that $$f$$ and $$K$$ are differentiable on this intervals. Right?

 

[HallsofIvy]

Yes, $[a, b]\times [a, b]$ is the "Cartesian product" (named for DesCartes so what you mean by "Dekart product"), the set of all ordered pairs of numbers from the interval [a, b].

However, C([a, b]) is NOT the set of differentiable functions. It means simply functions that are continuous on [a, b], not necessarily differentiable. C¹([a, b]) is the set of functions that are at least once differentiable on [a, b].

 

[matematikuvol]

Tnx for the answer.