Can a piecewise function of a single variable without indefinite limit can be always concentrate into a single variable function (without the heaviside step function)? Well I take an example :
Let $0.5<x\leq 1$ then define :
$$f(x)=\frac{1}{x}$$
Let $x\geq 1$ then define :
$$g(x)=x$$
Now we can concentrate the functions $f(x),g(x)$ into a new function I call $h(x)$ wich is ($x\in[0.5,\infty)$):
$$h(x)=0.5\left(x+\frac{1}{x}+\sqrt{\left(x+\frac{1}{x}\right)^{2}-4}\right)$$
I take a simple case but my intuition says that it's not always possible even if I haven't a counter-example .
Question :
Is it legitimate to think that my question above is necessarly false ?
Thanks !!