As the title, what single word should be filled in the blank in
Analytic for complex function is to _________ for real function.
?
My first thought is ‘differentiable’. However, then I think differentiability is weaker than analyticity.
Consider the function $f(x):\mathbb{R}\mapsto\mathbb{R}$ defined as $$f(x)=\begin{cases} \sin x, & x\le 0\\ x, & x>0\\ \end{cases}$$ This real function is differentiable everywhere in its domain.
However, analytic complex functions(entire functions) $g(z):\mathbb{C}\mapsto \mathbb{C}$ certainly do not allow piecewise definitions. Therefore, I think differentiability is a weaker condition.
Any idea?