Question

Let f (x) = | x | + | x − 1|, then
(a) f (x) is continuous at x = 0, as well as at x = 1
(b) f (x) is continuous at x = 0, but not at x = 1
(c) f (x) is continuous at x = 1, but not at x = 0
(d) none of these

Answer 1

(a) f (x) is continuous at x = 0, as well as at x = 1

Since modulus function is everywhere continuous , $\left|x\right|\mathrm{and}\left|x-1\right|$ are also everywhere continuous.

Also,
It is known that if f and g are continuous functions, then f + g will also be continuous.

Thus, ​ ​$\left|x\right|+\left|x-1\right|$ is everywhere continuous.

Hence, $f\left(x\right)$ is continuous at $x=0\mathrm{and}x=1$.