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