Continuity of absolute value of a function
Solution 1:
Not quite: In order to show that $|f|$ is continuous, one needs to estimate $\Big||f(x) - |f(c)|\Big|$, not just show that $|f(x)| - |f(c)|$ can be bounded above (how do we know that $|f(x)| - |f(c)|$ is not some really large negative number?). But happily, this is also a consequence of the reverse triangle inequality, which states in full that
$$\Big||a| - |b|\Big| \le |a - b|$$
Therefore, we can just say that
$$\Big||f(x)| - |f(c)|\Big| \le |f(x) - f(c)| \le \epsilon$$
for some appropriately constrained $x, c$.