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Summary

Description	English: Illustration that the signum function is not continuous at the point $x=0$ with $\epsilon ={\tfrac {1}{2}}$ and $\delta ={\tfrac {1}{2}}$ . The point $x={\tfrac {\delta }{2}}$ is marked red.
Date	9 March 2017
Source	Own work
Author	Stephan Kulla (User:Stephan Kulla)
Other versions	$ϵ = 1 2 {\displaystyle \epsilon ={\tfrac {1}{2}}} and δ = 1 2 {\displaystyle \delta ={\tfrac {1}{2}}}$ $\epsilon ={\tfrac {1}{2}}$ and $\delta ={\tfrac {1}{2}}$ $ϵ = 1 2 {\displaystyle \epsilon ={\tfrac {1}{2}}} and δ = 1 {\displaystyle \delta =1}$ $\epsilon ={\tfrac {1}{2}}$ and $\delta =1$ $ϵ = 1 2 {\displaystyle \epsilon ={\tfrac {1}{2}}} and δ = 1 2 {\displaystyle \delta ={\tfrac {1}{2}}} and marked x = δ 2 {\displaystyle x={\tfrac {\delta }{2}}}$ $\epsilon ={\tfrac {1}{2}}$ and $\delta ={\tfrac {1}{2}}$ and marked $x={\tfrac {\delta }{2}}$ $ϵ = 3 2 {\displaystyle \epsilon ={\tfrac {3}{2}}} and δ = 1 {\displaystyle \delta =1}$ $\epsilon ={\tfrac {3}{2}}$ and $\delta =1$
SVG development InfoField	The SVG code is valid. This plot was created with Matplotlib.Category:Valid SVG created with Matplotlib code#Illustration%20of%20discontinuity%20with%20sgn(x)%20(version%203).svg
Source code InfoField	Python code from matplotlib.patches import Rectangle import matplotlib.pyplot as plt import numpy as np bound=2 eps=0.5 delta=0.5 f, ax = plt.subplots() ax.axhline(color="black") ax.axvline(color="black") ax.set_xlim(-bound,bound) ax.set_ylim(-2,2) ax.add_patch( Rectangle((-delta,-eps), 2delta, 2eps, edgecolor="black", alpha=0.6, linestyle="solid", facecolor="#dddddd", lw=1.5)) ax.plot([-bound,0],[-1,-1], color="blue") ax.plot([0,bound],[1,1], color="blue") ax.plot([-delta,0],[-1,-1], color="red", lw=2) ax.plot([0,delta],[1,1], color="red", lw=2) ax.plot([delta/2], [1], "o", color="red", lw=2) ax.plot([0,0],[1,-1], "o", color="white") ax.plot([0], [0], "o", color="green") plt.savefig("Illustration of discontinuity with sgn(x) (version 3).svg")

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Category:Signum function Category:Continuity (functions)

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File:Illustration of discontinuity with sgn(x) (version 3).svg

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