Why does piecewise plot have a discontinuity when the function, first and second derivatives are equal?

Question

There were some related questions to this one, but in this case we have the function value, the first derivative and the second derivative all equal (that was actually the problem I was solving).

We have:

f[x_] := Piecewise[{{3 x^2 - 3 x + 1, x >= 1}, {x^3, x < 1}}]
Plot[f[x], {x, 0, 1.5}]

The answer in those other questions was to use "Exclusions -> None" and it can be used here, but why is this happening since we are good in all derivative information?

Update

Some examples of the related questions are:

• Why does Plot leave gaps in the graph of a continuous function?
• Plot showing discontinuity where it shouldn't
• Mathematica plots a discontinuity in piecewise function that does not exist

Answer 1

Mathematica even leaves a gap when the expressions in Piecewise are equal, as long as Mathematica doesn't see the equality. Very simple example

test[x_] := Piecewise[{{x, x >= 1}, {Sqrt[x^2], x < 1}}]
Plot[test[x], {x, 0, 2}, PlotStyle -> Thick]

When you replace Sqrt[x^2] by x, no gap.

What you have to understand is that the cracks are features when you use Piecewise because usually a piecewise function has jumps or discontinuities. Mathematica really just splits the plot, when it sees Piecewise and cannot determine a simple equality between the expressions.

This is a bit unfortunate, because although it is correct most of the time, users will always complain why this doesn't work like they want it.

My short answer: If you don't want cracks, then use Exclusions->None or make sure Mathematica doesn't see your Piecewise

f[x_?NumericQ] := Piecewise[{{3 x^2 - 3 x + 1, x >= 1}, {x^3, x < 1}}]
Plot[f[x], {x, 0, 1.5}]

Here, the NumericQ hinders Mathematica to evaluate your expression for non-numeric values and the only chance it has is to put a number in and get a number out :-)

Btw, let's give Mathematica something to think about and add a second definition for f where it sees the Piecewise. It doesn't matter what it sees, so screw it:

f[x_] := Piecewise[{{"Ding", x < .4}, {"Dong", x < .8}, {"Boing", x < 1.2}}, "Blub"];

The important part is that Mathematica uses the above definition to see a Piecewise and the gaps it introduces. For plotting, Mathematica has to supply real numbers into f and then our very first definition is used. I hope this explains, why the following plot looks as it looks:

Plot[f[x], {x, 0, 1.5}]