  We characterize the limiting behavior of partial sums of multiplicative
functions $f:\mathbb{F}_q[t]\to S^1$. In contrast to the number field setting,
the characterization depends crucially on whether the notion of discrepancy is
defined using long intervals, short intervals, or lexicographic intervals.
  Concerning the notion of short interval discrepancy, we show that a
completely multiplicative $f:\mathbb{F}_q[t]\to\{-1,+1\}$ with $q$ odd has
bounded short interval sums if and only if $f$ coincides with a "modified"
Dirichlet character to a prime power modulus. This confirms the function field
version of a conjecture over $\mathbb{Z}$ that such modified characters are
extremal with respect to the growth rate of partial sums.
  Regarding the lexicographic discrepancy, we prove that the discrepancy of a
completely multiplicative sequence is always infinite if we define it using a
natural lexicographic ordering of $\mathbb{F}_{q}[t]$. This answers a question
of Liu and Wooley.
  Concerning the long sum discrepancy, it was observed by the Polymath 5
collaboration that the Erd\H{o}s discrepancy problem admits infinitely many
completely multiplicative counterexamples on $\mathbb{F}_q[t]$. Nevertheless,
we are able to classify the counterexamples if we restrict to the class of
modified Dirichlet characters. In this setting, we determine the precise growth
rate of the discrepancy, which is still unknown for the analogous problem over
the integers.
