MHB Continuous Except Jump Discontinuity: Showing True for Any Function

[Dustinsfl]

Show that a function on (a,b) that is continuous except for a jump discontinuity at $x_0\in(a,b)$ is of the form
$$
f(x) = g(x) + cH(x - x_0)
$$
where c is a constant and g is continuous on (a,b) except possibly for a removable discontinuity at $x_0$.

I know that is true since this how I construct those functions but not sure how to show it is true for any arbitrary function.

 

[Ackbach]

Show that a function on (a,b) that is continuous except for a jump discontinuity at $x_0\in(a,b)$ is of the form
$$
f(x) = g(x) + cH(x - x_0)
$$
where c is a constant and g is continuous on (a,b) except possibly for a removable discontinuity at $x_0$.

I know that is true since this how I construct those functions but not sure how to show it is true for any arbitrary function.

Well, it's not any ol' arbitrary function. It's continuous except for one jump discontinuity. I think you could definitely say that
$$c= \lim_{x \to x_{0}^{+}}f(x)- \lim_{x \to x_{0}^{-}}f(x).$$
So, suppose you define $c$ this way, and then let
$$g(x) := f(x)-c H(x-x_{0}).$$
You must then prove that $g$ is continuous on $(a,b)$ except possibly at $x_{0}$. If you could do that, I think you'd be done, right?