Continuity 2

One-sided continuity

  • One-sided continuity is defined similarly to two-sided continuity.
    • Let \(a\) be a number and let \(f(x)\) be a function.
    • If we have \(\lim_{x\to a^-}f(x)=f(a)\), then we say that \(f(x)\) is continuous at \(a\) from the left.
    • If we have \(\lim_{x\to a^+}f(x)=f(a)\), then we say that \(f(x)\) is continuous at \(a\) from the right.
    • For example, consider the function \(f(x)=[[x]]\).
    • If \(a\) is not an integer, then we have \(\lim_{x\to a^-}f(x)=[[a]]=\lim_{x\to a^+}f(x)\), therefore \(f(x)\) is continuous at \(a\).
    • If \(a\) is an integer, then we have \(\lim_{x\to a^-}f(x)=[[a]]-1\) and \(\lim_{x\to a^+}f(x)=[[a]]\). Therefore, \(f(x)\) is continuous at \(a\) from the right.
    • Exercise. 2.5.42.

Continuity on an interval

  • Let \(f(x)\) be a function and \(I\) an interval. Then \(f(x)\) is continuous on \(I\), if it is continuous at each point of \(I\).
    • If an endpoint \(a\) of \(I\) is closed, and \(f(x)\) is only defined on the side of \(a\) in the direction of \(I\), then instead of requiring two-sided continuity at \(a\), we only require that \(f(x)\) is continuous at \(a\) from the direction of \(I\).
    • For example, let \(f(x)=1-\sqrt{1-x^2}\) and \(I=[-1,1]\).
    • For \(-1<a<1\), we require two-sided continuity: \[ \lim_{x\to a}f(x)=\lim_{x\to a}(1-\sqrt{1-x^2})=1-\lim_{x\to a}\sqrt{1-x^2}=1-\sqrt{1-a^2}=f(a). \]
    • Since the domain of \(f(x)=1-\sqrt{1-x^2}\) is \([-1,1]=I\), in case \(a=\pm1\), we only require continuity from the direction of \(I\): \[ \lim_{x\to(-1)^+}f(x)=1=f(-1),\quad\lim_{x\to1^-}f(x)=1=f(1). \]
    • All this shows that \(f(x)\) is continuous on \(I\).
    • Click here for a plot.
    • Note that the graph of \(f(x)\) is the lower half of the circle \[ x^2+(y-1)^2=1. \]
    • We say that \(f(x)\) is continuous everywhere, if it is continuous on the entire real line \(\mathbf R=(-\infty,\infty)\).
    • Exercises. 2.5: 26,46