# removable discontinuity

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##### 1: 1.4 Calculus of One Variable
A removable singularity of $f(x)$ at $x=c$ occurs when $f(c+)=f(c-)$ but $f(c)$ is undefined. … A simple discontinuity of $f(x)$ at $x=c$ occurs when $f(c+)$ and $f(c-)$ exist, but $f(c+)\not=f(c-)$. If $f(x)$ is continuous on an interval $I$ save for a finite number of simple discontinuities, then $f(x)$ is piecewise (or sectionally) continuous on $I$. For an example, see Figure 1.4.1
##### 2: Viewing DLMF Interactive 3D Graphics
Any installed VRML or X3D browser should be removed before installing a new one. …
##### 4: 10.25 Definitions
In particular, the principal branch of $I_{\nu}\left(z\right)$ is defined in a similar way: it corresponds to the principal value of $(\tfrac{1}{2}z)^{\nu}$, is analytic in $\mathbb{C}\setminus(-\infty,0]$, and two-valued and discontinuous on the cut $\operatorname{ph}z=\pm\pi$. … The principal branch corresponds to the principal value of the square root in (10.25.3), is analytic in $\mathbb{C}\setminus(-\infty,0]$, and two-valued and discontinuous on the cut $\operatorname{ph}z=\pm\pi$. …
##### 5: 1.10 Functions of a Complex Variable
This singularity is removable if $a_{n}=0$ for all $n<0$, and in this case the Laurent series becomes the Taylor series. …Lastly, if $a_{n}\not=0$ for infinitely many negative $n$, then $z_{0}$ is an isolated essential singularity. … An isolated singularity $z_{0}$ is always removable when $\lim_{z\to z_{0}}f(z)$ exists, for example $(\sin z)/z$ at $z=0$. … A cut domain is one from which the points on finitely many nonintersecting simple contours (§1.9(iii)) have been removed. … Branches of $F(z)$ can be defined, for example, in the cut plane $D$ obtained from $\mathbb{C}$ by removing the real axis from $1$ to $\infty$ and from $-1$ to $-\infty$; see Figure 1.10.1. …
##### 6: 25.14 Lerch’s Transcendent
25.14.1 ${\Phi\left(z,s,a\right)\equiv\sum_{n=0}^{\infty}\frac{z^{n}}{(a+n)^{s}}},$ $|z|<1$; $\Re s>1,|z|=1$.
##### 7: 26.15 Permutations: Matrix Notation
For $(j,k)\in B$, $B\setminus[j,k]$ denotes $B$ after removal of all elements of the form $(j,t)$ or $(t,k)$, $t=1,2,\ldots,n$. $B\setminus(j,k)$ denotes $B$ with the element $(j,k)$ removed. …
##### 8: Bibliography W
• R. Wong and Y.-Q. Zhao (1999a) Smoothing of Stokes’s discontinuity for the generalized Bessel function. II. Proc. Roy. Soc. London Ser. A 455, pp. 3065–3084.
• R. Wong and Y.-Q. Zhao (1999b) Smoothing of Stokes’s discontinuity for the generalized Bessel function. Proc. Roy. Soc. London Ser. A 455, pp. 1381–1400.
• ##### 9: 23.18 Modular Transformations
23.18.7 ${s(d,c)=\sum_{r=1}^{c-1}\frac{r}{c}\left(\frac{dr}{c}-\left\lfloor\frac{dr}{c}% \right\rfloor-\frac{1}{2}\right),}$ $c>0$.
##### 10: 10.2 Definitions
Except in the case of $J_{\pm n}\left(z\right)$, the principal branches of $J_{\nu}\left(z\right)$ and $Y_{\nu}\left(z\right)$ are two-valued and discontinuous on the cut $\operatorname{ph}z=\pm\pi$; compare §4.2(i). … The principal branches of ${H^{(1)}_{\nu}}\left(z\right)$ and ${H^{(2)}_{\nu}}\left(z\right)$ are two-valued and discontinuous on the cut $\operatorname{ph}z=\pm\pi$. …