Continuity of f(x) implies that |f(x) - f(c)| < epsilon whenever |x - c| < delta. Where "epsilon" and "delta" are positive real numbers.
Now,
| |f(x)| - |f(c)| | _{=}< |f(x) - f(c)| for all x, which completes the proof.
i.e. | |f(x)| - |f(c)| | _{=}< |f(x) - f(c)| < epsilon whenever |x - c| < delta, so |f(x)| is a continuous function.
Hope this helps.