Consider $f(x)=\left\{\begin{matrix} 1-x& 0
Is $f$ continuous at $0$ (or the other endpoints)? I tried verifying it with the epsilon delta argument when the limit is possible $1$ or $0$ and I got them both to agree...
My first conjecture was that if the limit was $1$ as x goes to $0$, I would choose $\delta = \epsilon$. If the limit was $0$ as $x\to0$, then I would choose $\delta = \epsilon+1$