So I define those two properties as ($\mathbb{R_H}$ denotes hyperreal numbers):
EVP: If $I$ is an interval and $f:I\rightarrow\mathbb{R_H}$, we say that $f$ has the extreme value property iff $f$ has its maximum value on that interval $I$. That is, $\forall a\leq b\in I$, $\exists x\in[a,b]\forall y\in[a,b]$ such that $f(y)\leq f(x)$.
IVP: If $I$ is an interval, and $f:I\rightarrow\mathbb{R_H}$, we say that $f$ has the intermediate value property iff whenever $a<b$ are points in $I$ and $f(a)< c<f(b)$, there is a $d$ between $a$ and $b$ such that $f(d)=c$.
My question is, in hyperreal set, does EVP imply IVP? How about the other way around? I suspect "no" for both of them but still cannot find a counter example. Cheers!