Note : I think i found the proper mathematical terms that corresponds to what exists in my country , if you notice an obvious error of international term , thanks to correct.
Context : Defining Riemann Integral for piecewise functions.
In most books , i can read the statement as followed : Let $f$ be a piecewise function on $\left[a,b\right]$.
$\forall \epsilon > 0 , \exists \theta$, a step function on $\left[a,b\right]$, such as $\vert f-\theta\vert\leq \epsilon$
For me , the last part of the sentence means , such as $\forall t\in \left[a,b\right],\vert f(t)-\theta(t)\vert\leq \epsilon $
I'm embarrassed because , for me , a piecewise function can have by definition some indefiniteness points , (on jumps) , as long as the definition that the left and right limit exists is respected.
I imagine we can decide a convention , let's say to define the function on a point of "jump" by the left or the right limit , but then , i guess it should be stated in the theorem.
Question : How to explain this ambiguity if it's one ?