  Let $(\Omega,{\cal F},P)$ be a probability space and $L^{0}({\cal F},R)$ the
algebra of equivalence classes of real-valued random variables on
$(\Omega,{\cal F},P)$. When $L^{0}({\cal F},R)$ is endowed with the topology of
convergence in probability, we prove an intermediate value theorem for a
continuous local function from $L^{0}({\cal F},R)$ to $L^{0}({\cal F},R)$. As
applications of this theorem, we first give several useful expressions for
modulus of random convexity, then we prove that a complete random normed module
$(S,\|\cdot\|)$ is random uniformly convex iff $L^{p}(S)$ is uniformly convex
for each fixed positive number $p$ such that $1<p<+\infty$.
