  We introduce Z-stability, a notion capturing the intuition that if a function
f maps a metric space into a normed space and if the norm of f(x) is small,
then x is close to a zero of f. Working in Bishop's constructive setting, we
first study pointwise versions of Z-stability and the related notion of good
behaviour for functions. We then present a recursive counterexample to the
classical argument for passing from pointwise Z-stability to a uniform version
on compact metric spaces. In order to effect this passage constructively, we
bring into play the positivity principle, equivalent to Brouwer's fan theorem
for detachable bars, and the limited anti-Specker property, an intuitionistic
counterpart to sequential compactness. The final section deals with connections
between the limited anti-Specker property, positivity properties, and
(potentially) Brouwer's fan theorem for detachable bars.
