gaps in plot of piecewise function

[Benjamin Hell]

Hi,
I want to plot a piecewise function, but I don't want any gaps to appear
at the junctures. An easy example is:

s[x_] := Piecewise[{{-Sqrt[2]/2*Sqrt[-x + 0.5] + 2, x < 0.5}, {2, x >=
0.5}}];
Plot[s[x], {x, 0, 1}]

It should be clear, that the piecewise function defined above is
continuous, even at x=0.5. So there should not be any gaps appearing in
the plot, but they do. Maybe it's a feature of mathematica, but
nevertheless I want to get rid of the gaps. Any suggestions on how to
achieve that.

Thanks in advance.

[David Park]

I'm not certain of the exact underlying mechanics, but basically because of
the steep curve as x -> 2 from below, and the piecewise function,
Mathematica sees a discontinuity and leaves a gap. The way to overcome this
is to use the Exclusions option.

s[x_] := Piecewise[{{-Sqrt[2]/2*Sqrt[-x + 0.5] + 2, x < 0.5}, {2,
x >= 0.5}}];

Plot[s[x], {x, 0, 1},
Exclusions -> None,
Frame -> True,
PlotRangePadding -> .1]

David Park
djm...@comcast.net
http://home.comcast.net/~djmpark/

[dh]

use the option:

On 10.03.2010 12:30, Benjamin Hell wrote:

> at the junctures. An easy example is:

> s[x_] := Piecewise[{{-Sqrt[2]/2*Sqrt[-x + 0.5] + 2, x< 0.5}, {2, x>=

E-Mail:<mailto:d...@metrohm.com>

Internet:<http://www.metrohm.com>

[Patrick Scheibe]

Hi,

short answer: use Which instead of Piecewise for plotting.

Long answer: I assume it's a hack which should provide that piecewise
defined functions are not connected since in cases of step-functions it
is usually wanted that plots are not connected:

step = Which[x < 0.5, 1, 0.5 < x < 1, 0.5, True, 0]
step2 = PiecewiseExpand[step]
Plot[#, {x, 0, 2}] & /@ {step, step2}

If you want to know a bit more detailed what happens in you example you
could compare the two plots with different settings for PlotPoints and
MaxRecursion:

s[x_] := Piecewise[{{-Sqrt[2]/2*Sqrt[-x + 1/2] + 2, x < 1/2}}, 2];
s2[x_] := Which[x < 1/2, -Sqrt[2]/2*Sqrt[-x + 1/2] + 2, True, 2];
Column[Manipulate[
Plot[#, {x, 0, 1}, MaxRecursion -> mr,
MeshStyle -> {Red, PointSize[0.005]}, Mesh -> All,
PlotPoints -> pp, ImageSize -> 500],
{{pp, 3, "PlotPoints"}, 3, 30, 1},
{{mr, 1, "MaxRecursion"}, 1, 10, 1}
] & /@ {s[x], s2[x]}]

If you look really closely you see that the Piecewise-stuff gets always
disconnected, no matter how many plotpoints you use. In real-life you
just don't see that there is a gap when you have enough plotpoints and a
moderate setting for maxrecursion.

Cheers
Patrick

On Wed, 2010-03-10 at 06:30 -0500, Benjamin Hell wrote:
> Hi,
> I want to plot a piecewise function, but I don't want any gaps to appear

> at the junctures. An easy example is:
>
> s[x_] := Piecewise[{{-Sqrt[2]/2*Sqrt[-x + 0.5] + 2, x < 0.5}, {2, x >=

[Matthias Hunstig]

Hi,

> I want to plot a piecewise function, but I don't want any gaps to appear
> at the junctures.

Try
Exclusions->None
as an option for Plot.

Regards,

Matthias

[Sjoerd C. de Vries]

Hi Benjamin,

You could increase MaxRecursion and/or MaxPoints

Plot[s[x], {x, 0, 1},MaxRecursion -> 15]

Cheers -- Sjoerd

[Peter Pein]

Try:

Plot[s[x], {x, 0, 1}, Exclusions -> None]

hth

[Raffy]

Plot[..., MaxRecursion -> 15]

[gekko]

It looks like the point x == 0.5 is being picked up as a possible
discontinuity and so the plot does not attempt to join at this point.
You can disable this using the Exclusions option:

Plot[s[x], {x,0,1}, Exclusions->None]

Cheers, P.

[dh]

Hi Beata,
you may e.g. use Reduce like:

lines = t /. {Reduce[{deriv2func == 0, Element[t, Reals]}, t] //
ToRules};
lines = Line[{{ #, -0.2}, {#, 0.2}}] & /@ lines;
Plot[{deriv2func}, {t, First[First[date]], First[Last[date]]},
Frame -> True, GridLines -> Automatic, Epilog -> {Red, lines}]

Daniel

--

Daniel Huber
Metrohm Ltd.
Oberdorfstr. 68
CH-9100 Herisau
Tel. +41 71 353 8585, Fax +41 71 353 8907

[Sjoerd C. de Vries]

The latter functions are equal for all x. Doesn't hold for the former two.

If you increase PlotPoints to well over a couple of hundred with
MaxRecursion at 15 I don't see a gap.

Cheers -- Sjoerd

> -----Original Message-----
> From: Patrick Scheibe [mailto:psch...@trm.uni-leipzig.de]
> Sent: 11 March 2010 14:23
> To: David Park; Benjamin Hell; Sjoerd C. de Vries; Peter Pein; gekko;
> Matthias Hunstig
> Cc: math...@smc.vnet.net
> Subject: Re: Re: gaps in plot of piecewise
> function
>
> Hi again,
>
> I hope everyone saw now that Exclusions->None or using not Piecewise
> but
> e.g. Which will do the trick. In the documentation it sounds to me that
> many functions are generally connected to Piecewise (look at Properties
> and Relations in the Piecewise doc).
>
> My question is, why would it be wrong to connect the plot in Piecewise
> when the Limits are the same? Following example:
>
> Manipulate[
> Plot[Piecewise[{{Exp[1] x, x < 1}}, Exp[x]], {x, 0, 3},

> MaxRecursion -> mr, MeshStyle -> {Red, PointSize[0.005]},
> Mesh -> All, PlotPoints -> pp,

> ImageSize -> 500], {{pp, 5, "PlotPoints"}, 3, 30,

> 1}, {{mr, 1, "MaxRecursion"}, 1, 10, 1}]
>

> The function has the same limit at x->1 and the same derivative. I
> would
> clearly expect a plot without a gap even without the Exclusions
> options.
> Where am I wrong?
>
> Is it too unpredictable to check at least numerically the limits?
> But why is this working?
>
> Manipulate[
> Plot[Piecewise[{{Sin[x], x < 1.334}}, Cos[ x - Pi/2]], {x, 0, 3},

> MaxRecursion -> mr, MeshStyle -> {Red, PointSize[0.005]},
> Mesh -> All, PlotPoints -> pp,

> ImageSize -> 500], {{pp, 5, "PlotPoints"}, 3, 30,

> 1}, {{mr, 1, "MaxRecursion"}, 1, 10, 1}]
>

> What bothers me is that when using PiecewiseExpand you get an
> equivalent
> presentation of one and the same function but you get different plots
> in
> an, say not really predictable way.
>
> Cheers
> Patrick

>
> On Thu, 2010-03-11 at 06:34 -0500, David Park wrote:
> > I'm not certain of the exact underlying mechanics, but basically
> because of
> > the steep curve as x -> 2 from below, and the piecewise function,
> > Mathematica sees a discontinuity and leaves a gap. The way to
> overcome this
> > is to use the Exclusions option.
> >

> > s[x_] := Piecewise[{{-Sqrt[2]/2*Sqrt[-x + 0.5] + 2, x < 0.5}, {2,
> > x >= 0.5}}];
> >

> > Plot[s[x], {x, 0, 1},
> > Exclusions -> None,
> > Frame -> True,
> > PlotRangePadding -> .1]
> >
> >
> > David Park
> > djm...@comcast.net
> > http://home.comcast.net/~djmpark/
> >
> >
> > From: Benjamin Hell [mailto:he...@exoneon.de]
> >

[Patrick Scheibe]

[Patrick Scheibe]

Hi,

> The latter functions are equal for all x. Doesn't hold for the former two.

yep, but

Sin[x] == (I/2)/E^(I*x) - (I/2)*E^(I*x) // Simplify

Manipulate[
Plot[Piecewise[{{Sin[x], x < 1.334}}, (I/2)/E^(I*x) - (I/2)*
E^(I*x)], {x, 0, 3}, MaxRecursion -> mr,

MeshStyle -> {Red, PointSize[0.005]}, Mesh -> All, PlotPoints -> pp,
ImageSize -> 500], {{pp, 5, "PlotPoints"}, 3, 30,
1}, {{mr, 1, "MaxRecursion"}, 1, 10, 1}]

> If you increase PlotPoints to well over a couple of hundred with

> MaxRecursion at 15 I don't see a gap.

"See" doesn't mean it's not there. Please set PlotPoints to 200 and
MaxRecursion to 15 and check the zoomed result

Manipulate[
Plot[Piecewise[{{Exp[1] x, x < 1}}, Exp[x]], {x, 0, 3},
MaxRecursion -> mr, MeshStyle -> {Red, PointSize[0.005]},

Mesh -> All, PlotPoints -> pp, ImageSize -> 500,
PlotRange -> {{1 - zoom, 1 + zoom}, Automatic}], {{pp, 5,
"PlotPoints"}, 3, 200, 1}, {{mr, 1, "MaxRecursion"}, 1, 15, 1},
{{zoom, 1}, 1, 0}]

Cheers
Patrick

> Cheers -- Sjoerd
>
> > -----Original Message-----
> > From: Patrick Scheibe [mailto:psch...@trm.uni-leipzig.de]
> > Sent: 11 March 2010 14:23
> > To: David Park; Benjamin Hell; Sjoerd C. de Vries; Peter Pein; gekko;
> > Matthias Hunstig
> > Cc: math...@smc.vnet.net
> > Subject: Re: Re: gaps in plot of piecewise
> > function
> >