Piecewise function does weird things when differentiated

Questions
1

First of all, when I plot the function like this, I get the seeming discontinuity, but when zooming in, it becomes smaller and smaller, suggesting it’s a visual bug (or so I assume, perhaps it’s on purpose), but when I then differentiate it, it gives me this, which I did not expect at all looking at the first graph.
What’s going on here? How can I get the “true” derivative, as it would be if the original function was continuous (and I thought it was, perhaps you guys can also tell whether or not this is indeed the case).

Thanks!

2

Plot[Piecewise[{{1 - Tan/2, 0 <= Tan && Tan < 1}, {Cot/2,1 <= Tan}}, 0], {x, 0, Pi/2}]

8332re16.png
8332re16.png798×563 29.2 KB

3

Hmmm,

I think the gap is due to the method how Mathematica produces plots, gathering values at different positions and refining and so.
The other question: Tan[ x ] = 1 means x = Pi / 4 , and for this x both constituents of your functions have the same values as well as their derivatives:

In[17]:= y1 = 1 - Tan[x]/2;
y2 = Cot[x]/2;
y1 /. x -> Pi/4
y2 /. x -> Pi/4

Out[19]= 1/2

Out[20]= 1/2



  (\!\(
   \*SubscriptBox[\(\[PartialD]\), \(x\)]\ y1\)) /. x -> Pi/4
   (\!\(
   \*SubscriptBox[\(\[PartialD]\), \(x\)]\ y2\)) /. x -> Pi/4
   
   -1
   
   -1

But the second derivatives are different, meaning the derivative of the composite function is not smooth. exactly as you see it. So it seems to me nothing is weird.

Look at

Plot[{y1, y2}, {x, 0, 1.5}, Plot[{y1, y2}, {x, 0, 1.5}, PlotStyle -> {Red, Blue}] 

Plot[Evaluate[{D[y1, x], D [y2, x]}], {x, 0, 1.5},  PlotStyle -> {Red, Blue}] 

Plot[Evaluate[{D[y1, x, x] , D[ y2, x, x]}], {x, 0, 1.5},  PlotStyle -> {Red, Blue}]