Corollary (Extreme Value Theorem): Let (X,d)(X,d)(X,d) be a compact metric space and let f:X→Rf:X\to\mathbb{R}f:X→R be continuous . Then there exist xmin⁡,xmax⁡∈Xx_{\min},x_{\max}\in Xxmin​,xmax​∈X such that f(xmin⁡)=HAHAHUGOSHORTCODE585s3HBHBx∈Xf(x),f(xmax⁡)=HAHAHUGOSHORTCODE585s4HBHBx∈Xf(x). f(x_{\min})=min _{x\in X} f(x),\qquad f(x_{\max})=max _{x\in X} f(x). f(xmin​)=HAHAHUGOSHORTCODE585s3HBHBx∈X​f(x),f(xmax​)=HAHAHUGOSHORTCODE585s4HBHBx∈X​f(x). ...

Corollary: Let (X,d)(X,d)(X,d) be a compact metric space and let f:X→Rf:X\to\mathbb{R}f:X→R be continuous . Then fff is bounded : there exists M<∞M<\inftyM<∞ such that ∣f(x)∣≤M|f(x)|\le M∣f(x)∣≤M for all x∈Xx\in Xx∈X. Connection to parent theorem: By the extreme value theorem , fff attains a maximum M+M_+M+​ and a minimum M−M_-M−​. Then ∣f(x)∣≤max⁡{∣M+∣,∣M−∣}|f(x)|\le \max\{|M_+|,|M_-|\}∣f(x)∣≤max{∣M+​∣,∣M−​∣}. ...

Corollary (Heine–Cantor): Let (X,dX)(X,d_X)(X,dX​) be a compact metric space and let (Y,dY)(Y,d_Y)(Y,dY​) be a metric space. If f:X→Yf:X\to Yf:X→Y is continuous , then fff is uniformly continuous on XXX. Connection to parent theorem: This is precisely the Heine–Cantor theorem .

Continuous functions are Riemann integrable: If f:[a,b]→Rf:[a,b]\to\mathbb{R}f:[a,b]→R is continuous , then fff is Riemann integrable on [a,b][a,b][a,b]. This theorem guarantees that the Riemann integral covers all standard functions from calculus and is a basic entry point for more refined integrability criteria. Proof sketch: A continuous function on the compact interval [a,b][a,b][a,b] is uniformly continuous . Given ε>0\varepsilon>0ε>0, choose δ>0\delta>0δ>0 so that ∣x−y∣<δ|x-y|<\delta∣x−y∣<δ implies ∣f(x)−f(y)∣<ε/(b−a)|f(x)-f(y)|<\varepsilon/(b-a)∣f(x)−f(y)∣<ε/(b−a). Take a partition PPP with mesh <δ<\delta<δ. Then on each subinterval, the oscillation of fff is <ε/(b−a)<\varepsilon/(b-a)<ε/(b−a), so U(f,P)−L(f,P)<εU(f,P)-L(f,P)<\varepsilonU(f,P)−L(f,P)<ε, proving integrability. ...

Corollary (Intermediate Value Theorem): Let I⊆RI\subseteq\mathbb{R}I⊆R be an interval and let f:I→Rf:I\to\mathbb{R}f:I→R be continuous . If a,b∈Ia,b\in Ia,b∈I and yyy lies between f(a)f(a)f(a) and f(b)f(b)f(b), then there exists ccc between aaa and bbb such that f(c)=y. f(c)=y. f(c)=y. Connection to parent theorem: This is the intermediate value theorem , which follows from connectedness of intervals and the fact that continuous images of connected sets are connected subsets of R\mathbb{R}R, hence intervals. ...

Let (X,d)(X,d)(X,d) be a metric space , let K⊆XK\subseteq XK⊆X be compact , and let f:K→Rkf:K\to\mathbb{R}^kf:K→Rk be continuous . Proposition: The image f(K)⊆Rkf(K)\subseteq\mathbb{R}^kf(K)⊆Rk is closed and bounded . This is the Euclidean specialization of “continuous image of compact is compact ” plus the Heine–Borel characterization of compact sets in Rk\mathbb{R}^kRk. ...

Let (X,d)(X,d)(X,d) be a metric space , let K⊆XK\subseteq XK⊆X be compact , and let f:K→Rf:K\to\mathbb{R}f:K→R be continuous . Proposition: The function fff is bounded on KKK: there exists M<∞M<\inftyM<∞ such that ∣f(x)∣≤M|f(x)|\le M∣f(x)∣≤M for all x∈Kx\in Kx∈K. This proposition is one of the core reasons compactness is the right hypothesis for global control from local continuity. ...

Continuous image of compact set is compact: Let (X,dX)(X,d_X)(X,dX​) and (Y,dY)(Y,d_Y)(Y,dY​) be metric spaces , let K⊆XK\subseteq XK⊆X be compact , and let f:X→Yf:X\to Yf:X→Y be continuous . Then f(K)⊆Yf(K)\subseteq Yf(K)⊆Y is compact. This is one of the most important permanence properties of compactness and is used to prove the extreme value theorem and many existence statements. ...

Continuous image of connected set is connected: Let (X,dX)(X,d_X)(X,dX​) and (Y,dY)(Y,d_Y)(Y,dY​) be metric spaces , let E⊆XE\subseteq XE⊆X be connected , and let f:X→Yf:X\to Yf:X→Y be continuous . Then f(E)⊆Yf(E)\subseteq Yf(E)⊆Y is connected. This theorem is a basic structural fact: continuous maps cannot “tear apart” connected sets. It implies, for example, that continuous real functions map intervals to intervals. ...

Let (X,d)(X,d)(X,d) be a metric space and let T:X→XT:X\to XT:X→X. The map TTT is a contraction mapping (or contraction) if there exists a constant c∈[0,1)c\in[0,1)c∈[0,1) such that for all x,y∈Xx,y\in Xx,y∈X, d(T(x),T(y))≤c d(x,y). d\bigl(T(x),T(y)\bigr)\le c\,d(x,y). d(T(x),T(y))≤cd(x,y). The constant ccc is called a contraction constant. A contraction is a special case of a Lipschitz map: TTT is Lipschitz with constant LLL if d(Tx,Ty)≤Ld(x,y)d(Tx,Ty)\le L d(x,y)d(Tx,Ty)≤Ld(x,y) for all x,yx,yx,y. Contractions are exactly Lipschitz maps with L<1L<1L<1. ...