Consider the function f R rightarrow R f x x x x notequalt

Consider the function f: R rightarrow R: f (x) = {|x| + x, x notequalto 0 2, x = 0. Use epsilon - delta definition to show that f is not continuous in its domain?

Solution

If f is differentiable at x0, then f is continuous at x0.(Theorem)

d/dx |x|+x=x/|x| + 1
it will not give finite values for x not equal to 0 as x might be 0.1 and in that case |x| will be 0 hence infinite values hence not differentiable

hence function is not continuous in its domain