Question

A function f(x) is said to be continuous at, if for any E> 0. there is. a = 0 such that
O x + xol<o=f(x) = f(xas
x xol SS3 f(x) - fixo)ce
O x-xol<&= f(x) + f(xol se
x= xol<e f(x) + f(x) <8​

Answer 1

Answer:

Math typing error in this question and is not completed.

Answer 2

It seems this is what you are looking for:

A function f(x) is said to be continuous at x=a, if for and any δ>0, there exists ε>0 such that -

a) if |x-a|<δ ⇒ |f(x)-f(a)|<ε

b) if |x-a|<δ ⇒ |f(x)-f(a)|>ε

Answer:

The Correct Answer would be a) if |x-a|<δ ⇒ |f(x)-f(a)|<ε. The correct answer would be (a) and let us see how this is true.

Step-by-step explanation:

According to Cauchy's Definition of Continuity, If a real-valued function that is f, is defined on an open interval I, then the function f is said to be continuous at x=a (a∈I) if there exists two small positive integers in δ>0 and ε>0, ε is dependent on δ, in the neighbourhood of (a+δ) and (a-δ) where δ<<<a, such that:

                             |x-a|<δ   ⇒    |f(x)-f(a)|<ε

Hence, it is the correct answer of the question.