A real function f is said to be continuous at x = c, where c is any point in the domain of f if
Since, h is a very small positive number. i.e. left hand limit as x → c (LHL) = right hand limit as x → c (RHL) = value of function at x = c.
A function is continuous at x = c if
The function is defined for all real numbers, so we need to comment about its continuity for all numbers in its domain
Function is changing its nature (or expression) at x = 0, so we need to check its continuity at x = 0 first.
f (0) = 0 [using equation 1]
LHL ≠ RHL ≠ f (0)
∴ Function is discontinuous at x = 0
c be any real number such that c > 0
=> f (x) is continuous everywhere for x > 0.
Let c be any real number such that c < 0
Therefore f (c) =
=> f (x) is continuous everywhere for x < 0.
Hence, f (x) is continuous for all Real numbers except zero that is discontinuous at x = 0.