Example LF 2.4: $m  = 0; b = 1: f (x) = 0*x + 1$

Draw a mapping diagram yourself or use the diagram created with GeoGebra to explore the diagram further.
Given a point / number, $x$, on the source line, there is a unique arrow  meeting the target line at the point / number, $1$, which corresponds to the linear function's value for $x$.

Notice that for each unit increase in $x$ there is no increase  or decrease in the  value of $f(x)$.
Thus there is a constant rate of change for $f$, precisely $0$,  the value of $m$.