values on the boundaries of a open interval

Question

Say we have a simple function $f(x)=x\cdot e^{-10x}$, we want to obtain the maximum value of it in an open interval $x \in (0.2, 1)$. Loosely speaking, the maximum point is at $x=0.2$ which is not within the range. But I want to use the result $f(0.2)$ in a higher level of application. How should I name it $f(0.2)$? I am thinking something like ``the upper bound of $f$ in the open interval is $f(0.2)$'' ? Or something more accurate?

Answer 1

The function does not have a maximum on the domain. Its supremum is $(0.2)e^{-2}$ and this value is not attained. [The function is strictly decreasing on $(0.2,1]$].