Find a specific D, f , and g, such that f(x) ≤ g(x) for all x ∈ D, but sup f(x) > inf g(x), x∈D

Question

I am unsure on how to answer this question, as the entire f function must be equal to or less than g, yet the upper limit of f must be higher than the lower limit of g. These 2 statements seem to counter one another, but I am thinking perhaps they could be the same function? Would f(x)=g(x) be a sufficient example? Thank you for any help!

Answer 1

The supremum and infimum do not have to coincide to the same $x$. $\sup f(x)$ just has to greater than or equal to all values of $f(x)$ and $\inf g(x)$ just has to be less than or equal to all values of $g(x)$. As long as there are two points $x_1$ and $x_2$ such that $f(x_1)>g(x_2)$, you'll have a $D$, $f$, and $g$ that satisfy those conditions.

Hagen von Eitzen's example $D=\{0,1\}, f(x)=g(x)=x$ and Krish's example $D=[0,1],f(x)=x^2,g(x)=x$ both work because $\sup f(x)=1>\inf g(x)=0$.

Answer 2

Try $D=\{0,1\}$, $f(x)=g(x)=x$.