Example CCD.DDS.AV.1:Functions that use the absolute value in the definition are commonly used to illustrate functions that are not differentiable at a single point (a singularity for the function).
Consider $f(x) = 2 |x-1| - 1$ at $x=1$ in the GeoGebra sketch below with both mapping diagram and graph to visualize the failure of the limit in the definition of the derivative.

Martin Flashman, Sept. 9, 2016, Created with GeoGebra

Notice how the points on the graph are paired with the points on the mapping diagram.

Check the box Show Second Pair to see the second pair of data used to estimate a possible numerical value for the derivative of $f$.
You can change  the values of $h$ with the slider.
Check the box Show Derivative... to see tangent line and focus point for all numbers $x_0 \ne 1$ by moving x_0 on the source axis.
You can explore similar examples by changing the functions in the input box, f(x) = ........   .