Showing that a function is continuous everywhere.

Question

$$f(x)=\lvert x^2 + 2x - 3\rvert$$

In the above function we can show continuity at a point by finding the left hand and right hand limits at that point! But how do we show that, this function is continuous at everywhere?

Answer 1

The composition of two functions that are continuous everywhere is continuous everywhere. Since $|\cdot|$ and $x^2+2x−3$ are continuous everywhere, and your $f(x)$ is the composition of $|\cdot|$ and $x^2+2x−3$, you can conclude.

Answer 2

Note that $$f(x)=\lvert x^2 + 2x - 3\rvert$$

is the composite function of the absolute value function and the polynomial function $$x^2 + 2x - 3$$ which are continuous everywhere.

Answer 3

By definition, a function is continuous "everywhere" (on its domain), if it is continuous at each point of the domain.

So, as you can show continuity of $f(x)=\lvert x^2 + 2x - 3\rvert$ at each point of its domain (using limits for example), you can conclude that your function is continuous "everywhere".