2.4 | Continuity and One-Sided Limits


Let \(f\) be a function defined on an open interval containing all values of \(x\) close to \(a\). Then \(f\) is continuous at \(a\) if
$$
\lim_{x \rightarrow a} f(x) = f(a)
$$
If \(f\) is defined for all values of \(x\) close to \(a\) but the equation above is not satisfied, then \(f\) is discontinuous at \(a\) or \(f\) has a discontinuity at \(a\).
Let \(f\) is defined for all values of \(x\) close to \(a\) with
$$
\lim_{x \rightarrow a^-} f(x) = L
$$
and
$$
\lim_{x \rightarrow a^-} f(x) = M.
$$
If \(L\) and \(M\) are finite and \(L \not = M\) then \(f\) has a jump discontinuity at \(a\).
Let \(f\) is defined for all values of \(x\) close to \(a\) with
$$
\lim_{x \rightarrow a} f(x) = L.
$$
If \(L\) is finite and \(L \not = f(a)\) then \(f\) has a removable discontinuity at \(a\).
Let \(f\) is defined for all values of \(x\) close to \(a\). If either
$$
\lim_{x \rightarrow a^-} f(x) = \pm \infty
$$
or
$$
\lim_{x \rightarrow a^+} f(x) = \pm \infty
$$
then \(f\) has an essential discontinuity at \(a\).
A function \(f\) is continuous from the right at \(a\) if
$$
\lim_{x \rightarrow a^+} f(x) = f(a)
$$
A function \(f\) is continuous from the left at \(a\) if
$$
\lim_{x \rightarrow a^-} f(x) = f(a)
$$
A function \(f\) is continuous on an open interval \((a,b)\) if it is continuous at every number in the interval. A function \(f\) is continuous on a closed interval \([a,b]\) if it is continuous on \((a,b)\) and is also continuous from the right of \(a\) and from the left of \(b\). A function \(f\) is continuous on a half-open interval \([a,b)\) or \((a,b]\) if it is continuous on \((a,b)\) and \(f\) is continuous from the right of \(a\) or \(f\) is continuous from the left of \(b\), respectively.
If the functions \(f\) and \(g\) are continuous at \(a\), then the following functions are also continuous at \(a\).

  • \(f \pm g\)
  • \(fg\)
  • \(cf\), where \(c\) is any constant
  • \(\frac{f}{g}\), if \(g(a) \not = 0\)
  • A polynomial function is continuous on \((-\infty, \infty)\).
  • A rational function is continuous on its domain.
The functions \(\sin x, \cos x, \tan x, \sec x, \csc x,\) and \(\cot x\) are continuous at every number in their respective domains.
If \(f\) is continuous on its domain, then \(f^{-1}\) is continuous on its domain. Also, the functions \(\sin^{-1} x, \cos^{-1} x, \tan^{-1} x, \sec^{-1} x, \csc^{-1} x, \cot^{-1} x, a^x,\) and \(\log_a x\) are continuous on their respective domains.
If the function \(f\) is continuous at \(L\), then
$$
\lim_{x \rightarrow a} f(g(x)) = f\left(\lim_{x \rightarrow a}g(x) \right)
$$
If the function \(g\) is continuous at \(a\) and the function \(f\) is continuous at \(f(a)\), then the composition \(f \circ g\) is continuous at \(a\).
The Intermediate Value Theorem: If \(f\) is continuous on a closed interval \([a,b]\) and \(M\) is any number between \(f(a)\) and \(f(b)\), inclusive, then there is at least one number \(c\) in \([a,b]\) such that \(f(c) = M\).
If \(f\) is a continuous function on a closed interval \([a,b]\) and \(f(a)\) and \(f(b)\) have opposite signs, then the equation \(f(x) = 0\) has at least one solution in the interval \((a,b)\) or, equivalently, the function \(f\) has at least one zero in the interval \((a,b)\).

E 2.4 Exercises