Question

Discuss the continuity and differentiability of function f(x)=|x|+|x-1| in the interval (1,2)

Answer 1

It is modulus function.
Such a function is continuous at all points and not differentiable only when the function is zero.

The function has 2 parts, |x| and |x +1 |

|x | is continuous at all points, non differentiable at x=0.

for |x+1|, it is continuous at all points and non differentiable when x+1 = 0, ie, x= 1.

x=0 doesnot lie in interval (1,2)
Also, x=1 doesn't lie in the interval, as in (1,2 ), by open brackets, we mean that both 1 and 2 are not included.

So,
The function f(x) in interval (1,2) doesn't include any points where the sub functions |x|or |1+x| are discontinuous.

So, as f(x) is sum of these two functions, it is also continuous and differetiable at all points in interval (2,3).

The function is continuous at all points and non differentiable only when it tends to minimum value, ie at x=0 and x=1.