Continuity of limit function

Question

State the three conditions satisfied by the function f(x)  if it is continuous at x = c using mathematical notation. From this, justify mathematically whether the function  g(x)=x² -1  is continuous at x=0

Answer 1

In calculus 1, continuity is often defined using limits.

We say that f is continuous at a if lim_x{\to a} f(x) = f(a).

This definition is often written in 3 'parts' to help students understand what the definition entails:

1. f(a) exists
2. lim_{x \to a} f(x) exists
3. The results in 1. and 2. are equal.

In the specific case of f(x) = x^2-1, we have:

1. f(0) = -1
2. lim_{x \to 0} f(x) = -1
3. 1. and 2 are equal

Hence, f is continuous at x=0.

Answer 2

In order for a function f to be continuous at a point c, the following three conditions must hold:

1. f(c) is defined (c is in the domain of f)
2. lim f(x) when x-> c exists
3. lim f(x) when x->c =f(c) (the value of f equals the limit of f at c)

In this case:

g(x)=x^2-1, x=0

1. g(0)=0^2-1=-1
2. lim g(x) when x->0=0^2-1=-1
3. lim g(x)=-1=g(0)

therefore all conditions are met so g(x) =x^2-1 is continuous at x=0