Q. 13 - Calculus

Question

State what it means for a function f to be continuous at a point x = c, in terms of the delta–epsilon definition of limit. 

Step-by-Step Solution

Answer

The function f to be continuous at a point x = c, in terms of the delta–epsilon definition of limit is $\underset{x\to c}{\lim }f\left(x\right)=L$ for all number $\epsilon >0,$ there exists some real number $\delta >0$ such that if $0<|x-c|<\delta , \mathrm{then} \left|f\right(x)-L|<\epsilon .$

Step 1. Given Information.

The function f is continuous at a point $x=c.$

Step 2. Stating.

The function f  to be continuous at a point x = c, in terms of the delta-epsilon definition of limit.  

Let f(x) be a function defined on the interval that contains x=c, then the limit $\underset{x\to c}{\lim }f\left(x\right)=L$ for all number $\epsilon >0,$ there exists some real number $\delta >0$ such that if $0<|x-c|<\delta , \mathrm{then} \left|f\right(x)-L|<\epsilon .$