# discontinuity

(0.001 seconds)

## 1—10 of 17 matching pages

##### 1: 1.4 Calculus of One Variable
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: 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 $\operatorname{ph}z=\pm\pi$. …
##### 3: 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.
• ##### 4: 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 $\operatorname{ph}z=\pm\pi$. …
##### 5: 4.15 Graphics Figure 4.15.4: arctan ⁡ x and arccot ⁡ x . … arccot ⁡ x is discontinuous at x = 0 . Magnify
##### 7: 4.2 Definitions
Consequently $\ln z$ is two-valued on the cut, and discontinuous across the cut. … This is an analytic function of $z$ on $\mathbb{C}\setminus(-\infty,0]$, and is two-valued and discontinuous on the cut shown in Figure 4.2.1, unless $a\in\mathbb{Z}$. …
##### 8: 28.12 Definitions and Basic Properties
As a function of $\nu$ with fixed $q$ ($\neq 0$), $\lambda_{\nu}\left(q\right)$ is discontinuous at $\nu=\pm 1,\pm 2,\dots$. …
##### 9: Bibliography D
• T. M. Dunster (1996b) Asymptotics of the generalized exponential integral, and error bounds in the uniform asymptotic smoothing of its Stokes discontinuities. Proc. Roy. Soc. London Ser. A 452, pp. 1351–1367.