3.3: Continuity
- Page ID
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)Let \(D\) be a nonempty subset of \(\mathbb{R}\) and let \(f: D \rightarrow \mathbb{R}\) be a function. The function \(f\) is said to be continuous at \(x_{0} \in D\) if for any real number \(\varepsilon > 0\), there exists \(\delta > 0\) such that if \(x \in D\) and \(\left|x-x_{0}\right|<\delta\), then
\[\left|f(x)-f\left(x_{0}\right)\right|<\varepsilon .\]
If \(f\) is continuous at every point \(x \in D\), we say that \(f\) is continuous on \(D\) (or just continuous if no confusion occurs).
Figure \(3.1\): Definition of Continuity.
Let \(f: \mathbb{R} \rightarrow \mathbb{R}\) be given by \(f(x)=3 x+7\).
Solution
Let \(x_{0} \in \mathbb{R}\) and let \(\varepsilon > 0\). Choose \(\delta=\varepsilon / 3\). Then if \(\left|x-x_{0}\right|<\delta\), we have
\[\left|f(x)-f\left(x_{0}\right)\right|=\left|3 x+7-\left(3 x_{0}+7\right)\right|=\left|3\left(x-x_{0}\right)\right|=3\left|x-x_{0}\right|<3 \delta=\varepsilon .\]
This shows that \(f\) is continuous at \(x_{0}\).
Note that the above definition of continuity does not mention limits. This allows to include in the definition, points \(x_{0} \in D\) which are not limit points of \(D\). If \(x_{0}\) is an isolated point of \(D\), then there is \(\delta > 0\) such that \(B\left(x_{0} ; \delta\right) \cap D=\left\{x_{0}\right\}\). It follows that for \(x \in B\left(x_{0} ; \delta\right) \cap D\), \(\left|f(x)-f\left(x_{0}\right)\right|=0<\varepsilon\) for any epsilon. Therefore, every function is continuous at an isolated point of its domain.
To study continuity at limit points of \(D\), we have the following theorem which follows directly from the definitions of continuity and limit.
Let \(f: D \rightarrow \mathbb{R}\) and let \(x_{0} \in D\) be a limit point of \(D\). Then \(f\) is continuous at \(x_{0}\) if and only if
\[\lim _{x \rightarrow x_{0}} f(x)=f\left(x_{0}\right).\]
- Proof
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Let \(f: \mathbb{R} \rightarrow \mathbb{R}\) be given by \(f(x)=3 x^{2}-2 x+1\).
Solution
Fix \(x_{0} \in \mathbb{R}\). Since, from the results of the previous theorem, we have
\[\lim _{x \rightarrow x_{0}} f(x)=\lim _{x \rightarrow x_{0}}\left(3 x^{2}-2 x+1\right)=3 x_{0}^{2}-2 x_{0}+1=f\left(x_{0}\right) \nonumber\]
it follows that \(f\) is continuous at \(x_{0}\).
The following theorem follows directly from the definition of continuity, Theorem 3.1.2 and Theorem 3.3.2 and we leave its proof as an exercise.
Let \(f: D \rightarrow \mathbb{R}\) and let \(x_{0} \in D\). Then \(f\) is continuous at \(x{0}\) if and only if for any sequence \(\left\{x_{k}\right\}\) in \(D\) that converges to \(x_{0}\), the sequence \(\left\{f\left(x_{k}\right)\right\}\) converges to \(f\left(x_{0}\right)\).
- Proof
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The proofs of the next two theorems are straightforward using Theorem 3.3.3.
Let \(f, g: D \rightarrow \mathbb{R}\) and let \(x_{0} \in D\). Suppose \(f\) and \(g\) are continuous at \(x_{0}\). Then
- \(f+g\) and \(fg\) are continuous at \(x{0}\).
- \(cf\) is continuous at \(x_{0}\) for any constant \(c\).
- If \(g\left(x_{0}\right) \neq 0\), then \(\frac{f}{g}\) (defined on \(\widetilde{D}=\{x \in D: g(x) \neq 0\}\)) is continuous at \(x_{0}\).
- Proof
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We prove (a) and leave the other parts as an exercise. We will use Theorem 3.3.3. Let \(\left\{x_{k}\right\}\) be a sequence in \(D\) that converges to \(x_{0}\). Since \(f\) and \(g\) are continuous at \(x_{0}\), by Theorem 3.3.3 we obtain that \(\left\{f\left(x_{k}\right)\right\}\) converges to \(f\left(x_{0}\right)\) and \(\left\{g\left(x_{k}\right)\right\}\) converges to \(g\left(x_{0}\right)\). By Theorem 2.2.1 (a), we get that \(\left\{f\left(x_{k}\right)+g\left(x_{k}\right)\right\}\) converges to \(f\left(x_{0}\right)+g\left(x_{0}\right)\). Therefore,
\[\lim _{k \rightarrow \infty}(f+g)\left(x_{k}\right)=\lim _{k \rightarrow \infty} f\left(x_{k}\right)+g\left(x_{k}\right)=f\left(x_{0}\right)+g\left(x_{0}\right)=(f+g)\left(x_{0}\right) .\]
Since \(\left\{x_{k}\right\}\) was arbitrary, using Theorem 3.3.3 again we conclude \(f+g\) is continuous at \(x_{0}\). \(\square\)
Let \(f: D \rightarrow \mathbb{R}\) and let \(g: E \rightarrow \mathbb{R}\) with \(f(D) \subset E\). If \(f\) is continuous at \(x_{0}\) ang \(g\) is continuous at \(f\left(x_{0}\right)\), then \(g \circ f\) is continuous at \(x_{0}\).
- Proof
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Exercise \(\PageIndex{1}\)
Prove, using definition 3.3.1, that each of the following functions is continuous on the given domain:
- \(f(x)=a x+b, a, b \in \mathbb{R}, \text { on } \mathbb{R}\).
- \(f(x)=x^{2}-3 \text { on } \mathbb{R}\).
- \(f(x)=\sqrt{x} \text { on }[0, \infty)\).
- \(f(x)=\frac{1}{x} \text { on } \mathbb{R} \backslash\{0\}\).
Exercise \(\PageIndex{2}\)
Determine the values of \(x\) at which each function is continuous. The domain of all the functions is \(\mathbb{R}\).
- \(f(x)=\left\{\begin{array}{ll} \left|\frac{\sin x}{x}\right|, & \text { if } x \neq 0 \text{;} \\ 1, & \text { if } x=0 \text{.} \end{array}\right.\)
- \(f(x)=\left\{\begin{array}{ll} \frac{\sin x}{|x|}, & \text { if } x \neq 0 \text{;} \\ 1, & \text { if } x=0 \text{.} \end{array}\right.\)
- \(f(x)=\left\{\begin{array}{ll} x \sin \frac{1}{x}, & \text { if } x \neq 0 \text{;} \\ 0, & \text { if } x=0 \text{.} \end{array}\right.\)
- \(f(x)=\left\{\begin{array}{ll} \cos \frac{\pi x}{2}, & \text { if }|x| \leq 1 \text{;} \\ |x-1|, & \text { if }|x|>1 \text{.} \end{array}\right.\)
- \(f(x)=\lim _{n \rightarrow \infty} \sin \frac{\pi}{2\left(1+x^{2 n}\right)}, \quad x \in \mathbb{R}\).
Exercise \(\PageIndex{3}\)
Let \(f: \mathbb{R} \rightarrow \mathbb{R}\) be the function given by
\[f(x)=\left\{\begin{array}{ll} x^{2}+a, & \text { if } x>2 \text{;} \\ a x-1, & \text { if } x \leq 2 \text{.} \end{array}\right .\]
Find the value of \(a\) such that \(f\) is continuous
Exercise \(\PageIndex{4}\)
Let \(f: D \rightarrow \mathbb{R}\) and let \(x_{0} \in D\). Prove that if \(f\) is continuous at \(x_{0}\), then \(|f|\) is continuous at this point. Is the converse true in general
Exercise \(\PageIndex{5}\)
Prove Theorem 3.3.3. (Hint: treat separately the cases when \(x_{0}\) is a limit point \(D\) and when it is not.
Exercise \(\PageIndex{6}\)
Prove parts (b) and (c) of Theorem 3.3.4
Exercise \(\PageIndex{7}\)
Prove Theorem 3.3.5
Exercise \(\PageIndex{8}\)
Explore the continuity of the function \(f\) in each case below.
- Let \(g, h:[0,1] \rightarrow \mathbb{R}\) be continuous functions and define
\[f(x)=\left\{\begin{array}{ll}
g(x), & \text { if } x \in \mathbb{Q} \cap[0,1] \text{;} \\
h(x), & \text { if } x \in \mathbb{Q}^{c} \cap[0,1] \text{.}
\end{array}\right .\]
Prove that if \(g(a)=h(a)\), for some \(a \in[0,1]\), then \(f\) is continuous at \(a\).
- Let \(f:[0,1] \rightarrow \mathbb{R}\) be the function given by
\[f(x)=\left\{\begin{array}{ll}
x, & \text { if } x \in \mathbb{Q} \cap[0,1] \text{;} \\
1-x, & \text { if } x \in \mathbb{Q}^{c} \cap[0,1] \text{.}
\end{array}\right .\]
Find all the points on \([0,1]\) at which the function is continuous
Exercise \(\PageIndex{9}\)
Consider the Thomae function on \((0,1]\) by
\[f(x)=\left\{\begin{array}{ll}
\frac{1}{q}, & \text { if } x=\frac{p}{q}, p, q \in \mathbb{N} \text{, where } p \text{ and } q \text { have no common factors;} \\
0, & \text { if } x \text { is irrational. }
\end{array}\right .\]
- Prove that for every \(\varepsilon > 0\), the set
\[A_{\varepsilon}=\{x \in(0,1]: f(x) \geq \varepsilon\}\]
is finite.
- Prove that \(f\) is continuous at every irrational point, and discontinuous at every rational point.
- Answer
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Exercise \(\PageIndex{10}\)
Consider \(k\) distinct points \(x_{1}, x_{2}, \ldots, x_{k} \in \mathbb{R}\), \(k \geq 1\). Find a function defined on \(\mathbb{R}\) that is continuous at each \(x_{i}\), \(i=1, \ldots, k\), and discontinuous at all other points.
- Answer
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Exercise \(\PageIndex{11}\)
Suppose that \(f, g\) are continuous functions on \(\mathbb{R}\) and \(f(x)=g(x)\) for all \(x \in \mathbb{Q}\). Prove that \(f(x)=g(x)\) for all \(x \in \mathbb{R}\).
- Answer
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