Why does the first derivative of a piecewise continuous function have discontinuities?

Question

I have this piecewise continuous function which is also continuously differentiable over time :

psi[t_] := Piecewise[{{(1 + t)^3 (-3 t^2 + t), -1 <= t <= 0},
                      {(1 - t)^3 (3 t^2 + t), 0 <= t <= 1}}];

Now, for starters, when I Plot it, a discontinuity appears. This can easily be solved with a simple Exclusions -> None option in the Plotcommand.

But then, when I calculate its first derivative over time using D, I obtain the following:

D[psi[t], t]

And then when I try to plot it :

• Is there something wrong with my original psi[t_]function? (Is it not continuous for Mathematica?)
• Why are the limits of definition of the first derivative modified?
• Why is the first derivative discontinuous?

Now, the easy solution would be to construct the first derivative using the results proposed by the D function and re-defining the definition domain... But I really want to understand this issue (if there is one).

Answer 1

One way to do this is to define the derivative function:

dPsi[t_] = D[psi[t], t]

which can then be plotted:

Plot[dPsi[t], {t, -3, 3}, PlotRange -> All]

The problem with your original formulation is that D[psi[t],t] does not evaluate to a function, it is instead d_t. The first derivative is not discontinuous, as a function, but it does have different definitions that correspond to the points where your Piecewise function changes.

Answer 2

Somewhat surprisingly, the easiest solution seems to have been overlooked:

Plot[psi'[t], {t, -3, 3}, PlotRange -> All]

As psi[t] has already been defined, it makes sense to use Derivative[] for producing the derivative.

Answer 3

I have solved this with a little hack:

Plot[D[psi[x], x] /. x->t, {t,-3, 3}].

This way I force Mathematica to first find the derivative and then substitute t into x and evaluate. However, this calculates the derivative for every t, so for more complex functions, it may become a performance issue.