continuity

A function is continuous at if its limit at that point equates to the function value. This definition also applies to multivariate function

A continuous function is continuous everywhere on its domain if

A function like discontinues at because it’s undefined there

Process

1. the function is defined at the point
2. the limit exists at the point
3. the limit of the function is the function value at the point

Quick check

If the function is a combination of

elementary function

an elementary function is a function of a single variable (typically real or complex)

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the algebraic limit laws imply it is continuous on its domain.

Application

Continuity implies Limit

If is continuous on the domain , with

Extreme Value Theorem

extreme value theorem

If is continuous on a closed interval , then such that is a local minimum and is a local maximum value of :

Sister theorem: intermediate value theorem

Application

In mathematical optimization algorithm, so we can guarantee to find a local minimum in finite number of steps. Especially in functions with asymptote

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