A strictly monotonic function is a real-valued function that is either a strictly
increasing function or a strictly
decreasing function. In the first case, 
implies 
for every pair 
in the domain; in the second
case, 
implies 
for every such pair.
A monotonic function allows equal values at distinct arguments, whereas a strictly monotonic
function does not. For example, 
is a strictly
increasing function on 
,
and 
is a strictly decreasing function.
A constant function is a monotonic
function but is not strictly monotonic on an interval
containing more than one point.
A differentiable function with 
throughout an interval is a strictly
increasing function, and one with 
throughout an interval
is a strictly decreasing function,
by the mean-value theorem. However, strict
monotonicity does not require a nonzero derivative
at every point. For example, 
has 
.
