Given function is
f(x) = |x| + |x - 1|
Function is also written as
Obviously, in given function we need to discuss the continuity and differentiability of the function f(x) at x = 0 or 1 only.
For continuity at x = 0
(i),(ii) and (iii)
\(\lim\limits_{x \to0^+}f(x) =\)\(\lim\limits_{x \to0^-}f(x) \) = f(0)
Hence, f(x) is continuous at x = 0.
For differentiability at x = 0
(iv) and (v) ⇒ RHD ≠ LHD at x = 0.
Hence, f(x) is not differentiable at x = 0 but continuous at x = 0.
Similarly, we can prove f(x) is not differentiable at x = 1 but continuous at x = 1 (Do yourself)
