Home/Differential Calculus/Continuity/DefinitionSection 3.1DefinitionFormulavizCopy LaTeXf is continuous at a⟺limx→af(x)=f(a)f\text{ is continuous at }a \quad\Longleftrightarrow\quad \lim_{x\to a}f(x)=f(a)f is continuous at a⟺x→alimf(x)=f(a)⟺\Longleftrightarrow⟺limx→af(x)=f(a)\lim_{x\to a}f(x)=f(a)limx→af(x)=f(a)View detail →FormulavizCopy LaTeXf is continuous at [a,b]⇒f attains every value between f(a) and f(b)f\text{ is continuous at }[a,b] \Rightarrow f\text{ attains every value between }f(a)\text{ and }f(b)f is continuous at [a,b]⇒f attains every value between f(a) and f(b)Intermediate value theorem (qualitative statement).View detail →
FormulavizCopy LaTeXf is continuous at a⟺limx→af(x)=f(a)f\text{ is continuous at }a \quad\Longleftrightarrow\quad \lim_{x\to a}f(x)=f(a)f is continuous at a⟺x→alimf(x)=f(a)⟺\Longleftrightarrow⟺limx→af(x)=f(a)\lim_{x\to a}f(x)=f(a)limx→af(x)=f(a)View detail →
FormulavizCopy LaTeXf is continuous at [a,b]⇒f attains every value between f(a) and f(b)f\text{ is continuous at }[a,b] \Rightarrow f\text{ attains every value between }f(a)\text{ and }f(b)f is continuous at [a,b]⇒f attains every value between f(a) and f(b)Intermediate value theorem (qualitative statement).View detail →