4.6 Functions Having a Value At Which There Is No Rate of Change at a
You cannot presume that a function has a rate of change at every moment. A
simple example is $f(x) = |x ‑ 2| + 1$. It’s graph has a cusp at
$x = 2$. The graph of f over any interval that contains
$x = 2$ in its interior (that is, 2 is not an endpoint) will
contain that cusp. The cusp will always keep the function from having a
constant rate of change over such intervals. The animation in Figure 4.7
Figure 4.6.1. A function cannot have a rate of change at any moment
where its graph has a cusp.
Figure 4.6.1 illustrates why the function $ y = \left\lvert x-2\right\rvert
+ 1$ cannot have a rate of change at the moment that $x = 2$. The
function’s rate of change changes abruptly at $x = 2$ no matter
how small an interval we make. Therefore it will never have a rate of change
that is essentially constant over an interval containing $x = 2$.
The rate of change will be -1 to the left of $x = 2$ and +1 to the
right of $x = 2$. Any interval containing $x = 2$ will
contain an interval over which the function has a rate of change of -1 and
an interval over which the function has a rate of change of +1. The function
will never have a constant rate change over an entire interval that contains
$x = 2$.