discont - Maple Help

discont

find the discontinuities of a function (or generalized function) over the reals

 Calling Sequence discont(f, x )

Parameters

 f - algebraic expression in x x - name

Description

 • discont returns a set of values where it is possible (not necessarily certain) that discontinuities occur.
 • This function returns all the discontinuity points over the reals. This includes the points where the function goes to plus or minus infinity. Note that this can include ranges in RealRange format.
 • Note that Dirac, though not a standard function, is considered to have a discontinuity when the argument is zero. This is because many algorithms in Maple treat all functions as pointwise defined, even if they are generalized functions.
 • Multiple discontinuities may be expressed with the aid of extra variables with the names _Zn~, _NNn~, and _Bn~. When these variables appear in the answer, the expression f has discontinuities for all integer assignments to the variables _Zn~, for all non-negative integer assignments to the variables _NNn~, and for all binary assignments to the variables _Bn~.
 If the solution incudes one of these variables, you can use about to find out information on the variable.

Examples

 > $\mathrm{discont}\left(\frac{1}{x},x\right)$
 $\left\{{0}\right\}$ (1)
 > $\mathrm{discont}\left(\mathrm{tan}\left(x\right),x\right)$
 $\left\{\frac{{1}}{{2}}{}{\mathrm{\pi }}{+}{\mathrm{\pi }}{}{\mathrm{_Z1~}}\right\}$ (2)
 > $\mathrm{discont}\left(\mathrm{round}\left(3x-\frac{1}{2}\right),x\right)$
 $\left\{\frac{{1}}{{3}}{+}\frac{{\mathrm{_Z2~}}}{{3}}\right\}$ (3)
 > $\mathrm{discont}\left(\mathrm{GAMMA}\left(\frac{x}{2}\right),x\right)$
 $\left\{{-}{2}{}{\mathrm{_NN1~}}\right\}$ (4)

Find out more about the variable in the solution

 > $\mathrm{about}\left(\mathrm{_NN1}\right)$
 Originally _NN1, renamed _NN1~:   is assumed to be: AndProp(integer,RealRange(0,infinity))
 > $\mathrm{discont}\left(\frac{\mathrm{arctan}\left(\frac{1\mathrm{tan}\left(2x\right)}{2}\right)}{{x}^{2}-1},x\right)$
 $\left\{{-1}{,}{1}{,}\frac{{1}}{{4}}{}{\mathrm{\pi }}{+}\frac{{1}}{{2}}{}{\mathrm{\pi }}{}{\mathrm{_Z3~}}\right\}$ (5)
 > $\mathrm{discont}\left(\mathrm{Dirac}\left(x-1\right),x\right)$
 $\left\{{1}\right\}$ (6)
 > $f≔\frac{1}{\mathrm{sin}\left(x\right)-\frac{1}{2}}$
 ${f}{≔}\frac{{1}}{{\mathrm{sin}}{}\left({x}\right){-}\frac{{1}}{{2}}}$ (7)
 > $\mathrm{discont}\left(f,x\right)$
 $\left\{\frac{{1}}{{6}}{}{\mathrm{\pi }}{+}{2}{}{\mathrm{\pi }}{}{\mathrm{_Z4~}}{,}\frac{{5}}{{6}}{}{\mathrm{\pi }}{+}{2}{}{\mathrm{\pi }}{}{\mathrm{_Z4~}}\right\}$ (8)

Evaluating the function where it is discontinuous will result in an error.

 > $\genfrac{}{}{0}{}{f}{\phantom{x=\frac{\mathrm{Pi}}{6}+\frac{2\mathrm{Pi}\cdot 10}{3}-6\mathrm{Pi}}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}|\phantom{\rule[-0.0ex]{0.1em}{0.0ex}}\genfrac{}{}{0}{}{\phantom{f}}{x=\frac{\mathrm{Pi}}{6}+\frac{2\mathrm{Pi}\cdot 10}{3}-6\mathrm{Pi}}$

Example where a range is returned:

 > $\mathrm{discont}\left(\mathrm{dilog}\left(-\sqrt{y}\right),y\right)$
 $\left\{{0}{,}\left[{0}{,}{\mathrm{\infty }}\right)\right\}$ (9)