Continuity¶
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Throughout, \(X\) and \(Y\) are metric spaces with metrics \(d_X\) and \(d_Y\), respectively.
Limit of a Function¶
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Definition (Limit of a Function): Let \(E \subset X\), \(f : E \to Y\) and \(p \in E^{\uparrow}\). If
\[\exists q \in Y.\ \forall \varepsilon > 0.\ \exists \delta > 0.\ \forall x \in E.\ \bigl(0 < d_X(x,p) < \delta \Longrightarrow d_Y(f(x),q) < \varepsilon\bigr),\]then the limit of \(f\) at \(p\) is \(\displaystyle \lim_{x\to p} f(x)=q\).
Tip
As you move close to a point, your shadow also moves close to some point in the shadow-verse.
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Theorem: Consider every sequence of points \(\{p_n\}\) in \(E\) converging towards \(p\) (with \(p_n \ne p\)). Then their images in \(Y\) under \(f\), \(p_n \mapsto f(p_n)\), also converge towards some \(q \in Y\) iff the limit of \(f\) at \(p\) is \(q\). Symbolically,
\[\lim_{x\to p} f(x)=q \iff \forall\{p_n\}\in E,\ p_n\ne p,\ \bigl(\lim_{n\to\infty} p_n=p \Longrightarrow \lim_{n\to\infty} f(p_n)=q\bigr).\]
Limits of Real and Complex Valued Functions¶
Let \(E \subset X\) be a metric space and \(f:E\to \mathbb{C}\) and \(g:E\to\mathbb{C}\) be two complex valued functions.
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Definition: Define
\(f+g:E\to\mathbb{C}\) as \((f+g)(x)=f(x)+g(x)\)
\(f-g:E\to\mathbb{C}\) as \((f-g)(x)=f(x)-g(x)\)
\(fg:E\to\mathbb{C}\) as \((fg)(x)=f(x)g(x)\)
\(f/g:E\to\mathbb{C}\) as \((f/g)(x)=f(x)/g(x)\) where \(g(x)\ne 0\)
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Theorem: Let \(p \in E^{\uparrow}\) and \(\displaystyle \lim_{x\to p} f(x)=s\) and \(\displaystyle \lim_{x\to p} g(x)=t\). Then
\(\displaystyle \lim_{x\to p} (f+g)(x)=s+t\)
\(\displaystyle \lim_{x\to p} (f-g)(x)=s-t\)
\(\displaystyle \lim_{x\to p} (fg)(x)=st\)
\(\displaystyle \lim_{x\to p} (f/g)(x)=s/t\) where \(t\ne 0\)
Limits of Euclidean Vector Valued Functions¶
Let \(E \subset X\) be a metric space and \(f:E\to\mathbb{R}^k\) and \(g:E\to\mathbb{R}^k\) be Euclidean vector valued functions.
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Definition: Define
\(f+g:E\to\mathbb{R}^k\) as \((f+g)(x)=f(x)+g(x)\)
\(f\cdot g:E\to\mathbb{R}\) as \((f\cdot g)(x)=f(x)\cdot g(x)\)
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Theorem: Let \(p \in E^{\uparrow}\) and \(\displaystyle \lim_{x\to p} f(x)=u\) and \(\displaystyle \lim_{x\to p} g(x)=v\). Then
\(\displaystyle \lim_{x\to p} (f+g)(x)=u+v\)
\(\displaystyle \lim_{x\to p} (f\cdot g)(x)=u\cdot v\)
Continuity¶
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Definition (Continuity): Let \(E \subset X\), \(f:E\to Y\) and \(p\in E\). If
\[\forall \varepsilon>0.\ \exists \delta>0.\ \forall x\in E.\ \bigl(d_X(x,p)<\delta \Longrightarrow d_Y(f(x),f(p))<\varepsilon\bigr),\]then \(f\) is continuous at \(p\).
Tip
As you move close to a point, your shadow moves close to the shadow of that point.
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Definition (Continuous on :math:`E`): \(f\) is continuous on \(E\) if it is continuous \(\forall p\in E\).
Tip
If you are neighbours, you would end up as neighbours.
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Theorem: If \(p\) is a limit point of \(E\), then \(f\) is continuous at \(p\) \(\iff \displaystyle \lim_{x\to p} f(x)=f(p)\).
Continuity of Function Composition¶
Let \(X,Y,Z\) be metric spaces. For \(E\subset X\), let \(f:E\to Y\) and \(g:f(E)\to Z\).
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Theorem: If \(f\) is continuous at \(p\in E\) and \(g\) is continuous at \(f(p)\), then \(g\circ f\) is continuous at \(p\).
Continuity of Functions on Open/Closed Sets¶
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Theorem: For \(f:X\to Y\), \(f\) is continuous iff for every open set \(V\in Y\), the inverse image \(f^{-1}(V)\) is open in \(X\).
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Corollary: In the above case, for every closed set \(C\in Y\), \(f^{-1}(C)\) is closed in \(X\).
Tip
Continuous maps preserve openness/closeness.
Continuity of Real and Complex Valued Functions¶
Let \(f:X\to\mathbb{C}\) and \(g:X\to\mathbb{C}\) be complex valued functions.
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Theorem: If \(f\) and \(g\) are continuous on \(X\), then \(f+g\), \(f-g\), \(fg\) and \(f/g\) are continuous on \(X\).
Continuity of Euclidean Vector Valued Functions¶
Let \(f_1:X\to\mathbb{R}, \cdots, f_k:X\to\mathbb{R}\) be real valued functions and let \(f:X\to\mathbb{R}^k\) be defined as \(f(x)=(f_1(x),\cdots,f_k(x))\) for \(x\in X\).
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Theorem: \(f\) is continuous \(\iff\) each of \(f_1,\cdots,f_k\) is continuous.
Theorem: If \(f:X\to\mathbb{R}^k\) and \(g:X\to\mathbb{R}^k\) are continuous, then \(f+g\) and \(f\cdot g\) are continuous.
Continuity and Compactness¶
Generic Functions on Compact Domain¶
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Theorem: Let \(f:X\to Y\) be a continuous map where \(X\) is compact. Then \(f(X)\) is compact.
Theorem: Let \(f:X\to Y\) be a continuous bijection where \(X\) is compact. Then the inverse map \(f^{-1}(f(x))=x\) for \(x\in X\) is a continuous mapping of \(Y\) onto \(X\).
Euclidean Vector Valued Functions on Compact Domain¶
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Definition (Bounded Function): \(f:E\to\mathbb{R}^k\) is bounded iff \(\exists M>0\) such that \(\forall x\in E,\ |f(x)|\le M\).
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Theorem: Let \(f:X\to\mathbb{R}^k\) be continuous where \(X\) is compact. Then (a) \(f(X)\) is closed and bounded and (b) \(f\) is bounded.
Real Valued Functions on Compact Domain¶
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Theorem: Let \(f:X\to\mathbb{R}\) be continuous where \(X\) is compact. Then \(\exists p,q\in X\) such that
\[f(p)=\sup_{x\in X} f(x) \qquad f(q)=\inf_{x\in X} f(x).\]
Uniform Continuity and Compactness¶
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Definition (Uniform Continuity): \(f:X\to Y\) is uniformly continuous on \(X\) iff
\[\forall \varepsilon>0.\ \exists \delta>0.\ \forall p,q\in X.\ d_X(p,q)<\delta \Longrightarrow d_Y(f(p),f(q))<\varepsilon.\]
Tip
There is a mandate on how far your neighbours can relocate from you.
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Theorem: If \(X\) is compact, then every continuous map \(f:X\to Y\) is uniformly continuous.
Lipschitz Continuity¶
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Definition (Lipschitz Continuity): \(f:X\to Y\) is Lipschitz continuous on \(X\) iff
\[\exists K>0.\ \forall p,q\in X.\ d_Y(f(p),f(q))\le K\cdot d_X(p,q)\]where \(K\in\mathbb{R}\) is the Lipschitz constant.
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Theorem: Let \(f:[a,b]\to\mathbb{R}\) be continuous on \([a,b]\), differentiable in \((a,b)\). If \(f'\) is bounded, then \(f\) is Lipschitz continuous.
Theorem: Every Lipschitz continuous function is uniformly continuous.
Tip
Having steeper and steeper derivative need not be an issue for uniform continuity, but it kills Lipschitz continuity.
Continuity and Connectedness¶
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Theorem: Let \(f:X\to Y\) be continuous. If \(E\subset X\) is connected, then \(f(E)\) is connected.
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Theorem (Intermediate Value Theorem): Let \(f:[a,b]\to\mathbb{R}\) be continuous. Then \(f\) attains all intermediate values between \(f(a)\) and \(f(b)\). If \(f(a)<f(b)\), then \(\forall c\) with \(f(a)<c<f(b)\) there exists \(x\in(a,b)\) such that \(f(x)=c\).
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Corollary (Bolzano’s theorem): If a continuous function has values of opposite sign inside an interval, then it has a root in that interval.
Theorem (Fixed Point Theorem): Let \(f:[a,b]\to[a,b]\) be continuous. Then \(\exists x\in[a,b]\) such that \(f(x)=x\).
Discontinuities¶
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Definition: If \(f:X\to Y\) is not continuous at some \(x\in X\), then \(f\) is discontinuous at \(x\).
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The following definitions and properties apply to functions defined on an interval \((a,b)\in\mathbb{R}\) or a segment \([a,b]\in\mathbb{R}\).
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Definition (Left-hand limit): Let \(f:(a,b)\to Y\). For \(x\in(a,b]\), let \(\{t_n\}\) be a sequence in \((a,x)\) such that
\[\lim_{n\to\infty} t_n=x \Longrightarrow \lim_{n\to\infty} f(t_n)=q.\]Then \(f(x-)=q\).
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Definition (Right-hand limit): Let \(f:(a,b)\to Y\). For \(x\in[a,b)\), let \(\{t_n\}\) be a sequence in \((x,b)\) such that
\[\lim_{n\to\infty} t_n=x \Longrightarrow \lim_{n\to\infty} f(t_n)=q.\]Then \(f(x+)=q\).
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Definition (Discontinuity of First Kind): If \(f:(a,b)\to Y\) is discontinuous at \(x\in(a,b)\) but \(f(x+)\) and \(f(x-)\) exist, then \(f\) has a discontinuity of first kind (simple discontinuity) at \(x\).
Definition (Discontinuity of Second Kind): If \(f:(a,b)\to Y\) is discontinuous at \(x\in(a,b)\) and the discontinuity is not simple, it has a discontinuity of second kind at \(x\).
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Definition: If \(f(x-)=f(x)\) we say that \(f\) is continuous from the left at \(x\).
Definition: If \(f(x+)=f(x)\) we say that \(f\) is continuous from the right at \(x\).
The following sections apply to real valued functions \(f:X\to\mathbb{R}\).
Monotonic Functions¶
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Definition (Monotonically Increasing): \(f:(a,b)\to\mathbb{R}\) is monotonically increasing on \((a,b)\) if
\[a<x<y<b \Longrightarrow f(x)\le f(y).\]
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Theorem: Let \(f:(a,b)\to\mathbb{R}\) be monotonically increasing. Then \(\forall x\in(a,b)\), \(f(x+)\) and \(f(x-)\) exist and
\[\sup_{a<t<x} f(t)=f(x-)\le f(x)\le f(x+)=\inf_{x<t<b} f(t).\]Also, \(a<x<y<b \Longrightarrow f(x+)\le f(y-)\).
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Theorem: Let \(f:(a,b)\to\mathbb{R}\) be monotonic. The set of points in \((a,b)\) at which \(f\) is discontinuous is at most countable.
Infinite Limits and Limits at Infinity¶
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Definition: The neighbourhood of \(x\) is any segment \((x-\delta,x+\delta)\).
Definition: For \(c\in\mathbb{R}\), the set \((c,+\infty):=\{x\mid x>c\}\) is the neighbourhood of \(+\infty\) and \((-\infty,c):=\{x\mid x<c\}\) is the neighbourhood of \(-\infty\).
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Definition: Let \(E\subset \mathbb{R}\). For \(f:E\to \mathbb{R}\cup\{+\infty,-\infty\}\), \(\displaystyle \lim_{x\to p} f(x)=q\) iff
\[\forall N_Y(q).\ \exists N_X(p).\ x\in N_X^{*}(p)\cap E \Longrightarrow f(x)\in N_Y(q)\]where \(N_X(p)\cap E\ne\varnothing\).
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Theorem: Let \(f\) and \(g\) be defined on \(E\subset \mathbb{R}\) and \(\displaystyle \lim_{t\to x} f(t)=A\) and \(\displaystyle \lim_{t\to x} g(t)=B\).
\(f(t)\to A' \Longrightarrow A'=A\)
\((f+g)(t)\to A+B\)
\((fg)(t)\to AB\)
\((f/g)(t)\to A/B\).