  A celebrated theorem of Friedgut says that every function $f:\{0,1\}^n \to
\{0,1\}$ can be approximated by a function
  $g:\{0,1\}^n \to \{0,1\}$ with $\|f-g\|_2^2 \le \epsilon$ which depends only
on $e^{O(I_f/\epsilon)}$ variables where $I_f$ is the sum of the influences of
the variables of $f$. Dinur and Friedgut later showed that this statement also
holds if we replace the discrete domain $\{0,1\}^n$ with the continuous domain
$[0,1]^n$, under the extra assumption that $f$ is increasing. They conjectured
that the condition of monotonicity is unnecessary and can be removed.
  We show that certain constant-depth decision trees provide counter-examples
to Dinur-Friedgut conjecture. This suggests a reformulation of the conjecture
in which the function $g:[0,1]^n \to \{0,1\}$ instead of depending on a small
number of variables has a decision tree of small depth. In fact we prove this
reformulation by showing that the depth of the decision tree of $g$ can be
bounded by $e^{O(I_f/\epsilon^2)}$.
  Furthermore we consider a second notion of the influence of a variable, and
study the functions that have bounded total influence in this sense. We use a
theorem of Bourgain to show that these functions have certain properties. We
also study the relation between the two different notions of influence.
