Considering the following definition of continuity (there is nothing unusual yet here):
$$\forall \varepsilon > 0\ \exists \delta > 0\ \text{s.t. } 0 < |x - x_0| < \delta \implies |f(x) - f(x_0)| < \varepsilon $$
I always thought $\delta$, $\epsilon$ $\rightarrow$ 0.
However I was told today (and my assignment wash downgraded accordingly) that $\delta$ and $\epsilon$ are not necessary infinitesimally small numbers.
Can someone please share an insight into this idea and provide a detailed explanation on how I can show that a piece wise continuous function contains discontinuities without resorting to infinitesimally small $\delta$ and $\epsilon$ in the above continuity definition.
I am really struggling with this new removal of restriction on $\delta$ and $\epsilon$, so a really detailed explanation would be much appreciated.