3: Limits and Continuity
- Page ID
- 49112
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)In this chapter, we extend our analysis of limit processes to functions and give the precise definition of continuous function. We derive rigorously two fundamental theorems about continuous functions: the extreme value theorem and the intermediate value theorem.
- 3.1: Limits of Functions
- This page covers the concept of limits in functions, providing formal definitions and examples. It establishes the uniqueness of limits at limit points and describes conditions for their existence, citing the Dirichlet function as a case of non-existence. Additionally, exercises challenge students to prove specific limits and explore properties of limits in various contexts, reinforcing their understanding through practical applications.
- 3.2: Limit Theorems
- This page covers theorems and properties related to the limits of functions, focusing on operations such as addition and multiplication. It discusses convergence of sequences, one-sided limits, and conditions for limit existence using the delta-epsilon approach. The page also addresses properties for increasing functions and infinite limits, while examples illustrate practical applications.
- 3.3: Continuity
- This page defines the continuity of functions on real number subsets, explaining that a function is continuous at a point if small changes in input lead to small changes in output. Continuity throughout a domain is explored, alongside theorems related to limit points and function combinations. Additionally, the page includes exercises analyzing continuity in piecewise functions, emphasizing proof techniques and implications for absolute values and rational versus irrational inputs.
- 3.4: Properties of Continuous Functions
- This page covers key concepts in real analysis, focusing on compact subsets of \(\mathbb{R}\) and the implications for continuous functions. It details theorems that state continuous functions on compact sets yield compact, closed, and bounded images, highlighting the Extreme Value Theorem. The Intermediate Value Theorem (IVT) is explained, demonstrating its application in finding roots.
- 3.5: Uniform Continuity
- This page covers uniformly continuous functions, defining them and providing examples such as constant and linear functions. It explains Hölder continuity and its implication for uniform continuity. It also highlights that while all uniformly continuous functions are continuous, the reverse is not true, illustrated by the function \(f(x) = \frac{1}{x}\).
- 3.6: Limit Superior and Limit Inferior of Functions
- This page covers limit superior and limit inferior for real-valued functions, emphasizing their reliance on the function's behavior near limit points. It outlines mathematical definitions, conditions for establishing limit superior, and theorems linking these concepts to converging sequences. Important relationships are demonstrated, showing that convergence at a limit point occurs if the limit superior and inferior are equal.
- 3.7: Lower Semicontinuity and Upper Semicontinuity
- This page covers semicontinuity in functions, defining lower semicontinuity (l.s.c.) and upper semicontinuity (u.s.c.) based on their bounds in neighborhoods. Theorems are presented to characterize these types using limits of sequences. It highlights that l.s.c. functions on compact sets achieve absolute minima, and that the set where functions are less than or equal to a constant is closed for l.s.c. Conversely, closed sets imply l.s.c., with similar principles for u.s.c.