Q. 7 - Calculus

Question

what it means, in terms of limits, for a function f to be continuous at a point$x=c$, left continuous at$x=c$, and right continuous at$x=c.$ 

Step-by-Step Solution

Answer

If limit$\underset{x\to c}{\lim }f\left(x\right)=f\left(c\right),$ then function f is continuous at $x=c.$

If left limit $\underset{x\to c^{-}}{\lim }f\left(x\right)=f\left(c\right),$then function f is left continuous at $x=c.$

If Right limit $\underset{x\to c^{+}}{\lim }f\left(x\right)=f\left(c\right),$then function f is right continuous at $x=c.$

Step 1. Given information.

A function f is continuous at $x=c.$

function f is left continuous at $x=c.$

function f is right continuous at $x=c.$

Step 2. continuous function

A function is continuous at $x=c$ if its limit as $x\to c$ is equal to the function value at $x=c.$

$\underset{x\to c}{\lim }f\left(x\right)=f\left(c\right)$

Step 3. Left continuous function

A function is left continuous at $x=c$ if its left limit as $x\to c^{-}$ is equal to the function value at $x=c$.

$\underset{x\to c^{-}}{\lim }f\left(x\right)=f\left(c\right)$

Step 4. Right continuous function

A function is right continuous at $x=c$ if its right limit as $x\to c^{-}$ is equal to the function value at $x=c.$

$\underset{x\to c^{+}}{\lim }f\left(x\right)=f\left(c\right)$