Continuous

$\large f(x) \begin{cases}= x , {\text{ when }} x \ge 0 \\ = -x , {\text{when }} x<0 \end{cases} \\ {\text{So, only doubtful point for continuity of }} f(x) {\text{ is }} x = 0. \\ So, f(0) = 0 \\ {\text{Left hand limit}} = {\displaystyle \lim_{h \rightarrow 0}} f(0 - h) = {\displaystyle \lim_{h \rightarrow 0}} -(0 - h) = 0 \\ {\text{Right hand limit}} = {\displaystyle{\lim_{h \rightarrow 0}}} f(0 + h) = {\displaystyle{\lim_{h \rightarrow 0} }} (0 + h) = 0. \\ {\text{Since, }} f(0) = L.H.L = R.H.L.\\ {\text{So, function is continuous at }} x = 0 \\ \therefore f(x) {\text{ is }} {\boxed{{\text{continuous everywhere}}}}$

Here's a graph of this function:

The definition of continuity is a graphing that has no jumps, care to say. You should be able to draw a continuous graphing without lifting the pencil. This is true for this graphing - here's the numerical definition for that.

if We know that the function is continuous at $x=a$ . We test for the points, which give this to be true. As you approach any point on the function, from the left or right, either case is equal to the actual point on the line.