Sometimes, searching for formulas can be frustrating, hence, having a formula sheet helps you in solving problem easily. Instead of searching a single formula, you can easily look up formulas written on a separate page. That is why we came up with this idea to save your time and effort. Below are the formulas related to solving the continuity of functions.

Continuous Function at a Point

\exists f(a)

\exists \lim_{ x \rightarrow a } f(x) \Leftrightarrow \lim_{ x \rightarrow { a }^{ - } } = \lim_{ x \rightarrow { a }^{ + } } } f(x)

f(a) = \lim_{ x \rightarrow a } f(x)

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Directional Continuity

Left-continuous function

f(a) = \lim_{ x \rightarrow { a} ^{ - } } f(x)

Right-continuous function

f(a) = \lim_{ x \rightarrow { a }^{ + } } f(x)

 

Discontinuity

Removable Discontinuity

\nexists f(a) \quad or \quad f(a) \neq \lim_{ x \rightarrow a } f(x)

Jump Discontinuity

\lim_{ x \rightarrow { a }^{ - } } f(x) \neq \lim_{ x \rightarrow { a }^{ + } } f(x)

Essential Discontinuity

\nexists \lim_{ x \rightarrow { a }^{ - } } \quad or \quad \nexists \lim_{ x \rightarrow { a }^{ + } }

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Hamza

Hi! I am Hamza and I am from Pakistan. My hobbies are reading, writing and playing chess. Currently, I am a student enrolled in the Chemical Engineering Bachelor program.