Is the Set Defined by a Continuous Almost Everywhere Function Rectifiable?

[johnson12]

let g:[a,b] -> R be a function that is continuous almost everywhere. assume that g(x) > 0 on [a,b]. Show that the set
S = { (x,y): 0 <= y <= g(x) , a <= x <= b} is rectifiable.

One way to attack it, is to show that S is bounded and boundary of S has measure zero. the problem I am having is how to show that S is bounded, since g is continuous a.e. I don't now whether or not g is bounded on [a,b].

any comments at all are strongly appreciated, thanks.

 

[johnson12]

forgot to mention the definition of rectifiable here: a (bounded) set S is rectifiable if

$$\int_{S} 1$$ exists. (so it has volume.)update: PROBLEM HAS BEEN SOLVED.