Home/Differential Calculus/Continuity/Theorems on continuous functionsSection 3.3Theorems on continuous functionsFormulaCopy LaTeXf,g are continuous at a⇒f±g, fg, fg are continuous at a (g(a)≠0)f,g\text{ are continuous at }a \Rightarrow f\pm g,\ fg,\ \frac{f}{g}\text{ are continuous at }a\ (g(a)\neq0)f,g are continuous at a⇒f±g, fg, gf are continuous at a (g(a)=0)View detail →FormulaCopy LaTeXf is continuous at [a,b]⇒f attains absolute maximum and minimum on [a,b]f\text{ is continuous at }[a,b] \Rightarrow f\text{ attains absolute maximum and minimum on }[a,b]f is continuous at [a,b]⇒f attains absolute maximum and minimum on [a,b]Extreme value theorem.View detail →FormulaCopy LaTeXf is continuous at [a,b], f(a)≠f(b)⇒∀k∈(min{f(a),f(b)},max{f(a),f(b)}) ∃c∈(a,b):f(c)=kf\text{ is continuous at }[a,b],\ f(a)\neq f(b) \Rightarrow \forall k\in(\min\{f(a),f(b)\},\max\{f(a),f(b)\})\ \exists c\in(a,b): f(c)=kf is continuous at [a,b], f(a)=f(b)⇒∀k∈(min{f(a),f(b)},max{f(a),f(b)}) ∃c∈(a,b):f(c)=kFor every kkk between f(a)f(a)f(a) and f(b)f(b)f(b) (IVT).View detail →
FormulaCopy LaTeXf,g are continuous at a⇒f±g, fg, fg are continuous at a (g(a)≠0)f,g\text{ are continuous at }a \Rightarrow f\pm g,\ fg,\ \frac{f}{g}\text{ are continuous at }a\ (g(a)\neq0)f,g are continuous at a⇒f±g, fg, gf are continuous at a (g(a)=0)View detail →
FormulaCopy LaTeXf is continuous at [a,b]⇒f attains absolute maximum and minimum on [a,b]f\text{ is continuous at }[a,b] \Rightarrow f\text{ attains absolute maximum and minimum on }[a,b]f is continuous at [a,b]⇒f attains absolute maximum and minimum on [a,b]Extreme value theorem.View detail →
FormulaCopy LaTeXf is continuous at [a,b], f(a)≠f(b)⇒∀k∈(min{f(a),f(b)},max{f(a),f(b)}) ∃c∈(a,b):f(c)=kf\text{ is continuous at }[a,b],\ f(a)\neq f(b) \Rightarrow \forall k\in(\min\{f(a),f(b)\},\max\{f(a),f(b)\})\ \exists c\in(a,b): f(c)=kf is continuous at [a,b], f(a)=f(b)⇒∀k∈(min{f(a),f(b)},max{f(a),f(b)}) ∃c∈(a,b):f(c)=kFor every kkk between f(a)f(a)f(a) and f(b)f(b)f(b) (IVT).View detail →