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 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~.

Examples

 > $\mathrm{discont}\left(\frac{1}{x},x\right)$
 $\left\{{0}\right\}$ (1)
 > $\mathrm{discont}\left(\mathrm{tan}\left(x\right),x\right)$
 $\left\{{\mathrm{π}}{}{\mathrm{_Z1~}}{+}\frac{{1}}{{2}}{}{\mathrm{π}}\right\}$ (2)
 > $\mathrm{discont}\left(\mathrm{round}\left(3x-\frac{1}{2}\right),x\right)$
 $\left\{\frac{{1}}{{3}}{+}\frac{{1}}{{3}}{}{\mathrm{_Z2~}}\right\}$ (3)
 > $\mathrm{discont}\left(\mathrm{Γ}\left(\frac{x}{2}\right),x\right)$
 $\left\{{-}{2}{}{\mathrm{_NN1~}}\right\}$ (4)
 > $\mathrm{discont}\left(\frac{\mathrm{arctan}\left(\frac{1\mathrm{tan}\left(2x\right)}{2}\right)}{{x}^{2}-1},x\right)$
 $\left\{{-}{1}{,}{1}{,}\frac{{1}}{{2}}{}{\mathrm{π}}{}{\mathrm{_Z3~}}{+}\frac{{1}}{{4}}{}{\mathrm{π}}\right\}$ (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)
 > $\mathrm{discont}\left(f,x\right)$
 $\left\{\frac{{1}}{{6}}{}{\mathrm{π}}{+}{2}{}{\mathrm{π}}{}{\mathrm{_Z4~}}{,}\frac{{5}}{{6}}{}{\mathrm{π}}{+}{2}{}{\mathrm{π}}{}{\mathrm{_Z4~}}\right\}$ (8)

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

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