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
Solution:
(a) $f(x)$ is continuous at $x=0$, as well as at $x=1$
Since modulus function is everywhere continuous, $|x|$ and $|x-1|$ are also everywhere continuous.
Also,
It is known that if $f$ and $g$ are continuous functions, then $f+g$ will also be continuous.
Thus, $|x|+|x-1|$ is everywhere continuous.
Hence, $f(x)$ is continuous at $x=0$ and $x=1$.
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