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Definition of "Continuous Function"

Recall the definition of a limit:  \[\lim_{x\to c} f(x) = L\] means given any positive real number, ε, there exists a positive real number, δ, such that \[ 0 < |x-c| < \delta \Rightarrow |f(x) - L| < \varepsilon \] 

A function, f, is continuous on the open interval (a,b) means \[\lim_{x\to c} f(x) = f(c)\] for all c in (a,b).

A function, f, is continuous on the closed interval [a,b] means f is continuous on (a,b) and \[\lim_{x\to a+} f(x) = f(a)\],   and \[\lim_{x\to b-} f(x) = f(b)\].

Related pages in this website

Calculus Theorems -- important facts about continuous functions


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