Why does $(a, f(a))$ in this graph below not have a tangent line?

Question

Why does $(a, f(a))$ in this graph below not have a tangent line?

Thanks a lot!

Answer 1

Let's try an every-day analogy. When you are rolling a (prismatic) object on a plane (your table), the line along which you roll is the tangent at the contact point. See blue object below.

Now if at some point your object "makes an angle" (i.e, has a kink as shown in your snapshot, red object), then when that point touches the table, several inclinations are possible. This means that you do not have a unique line that can be called the tangent to the object at that point. In fact, you'll find that if you stop right when reaching that point as contact point, the inclination will depend on whether you approach from right or left. This illustrates the fact that there is a discontinuity in the tangent vector at that point.

Note: here I am asking you to visualize a 2D problem (tangent to a curve) with an everyday analogy, so a 3D one. For this to work, you have to be able to discard the direction perpendicular to the rolling one, so to have a prismatic object and consider just a vertical slice through it.

Note 2: my everyday analogue works only with convex shapes, for purely physical reasons (physics applies in addition to maths in everyday life!)