Functions continuous at a point

Question

Prove that if $ f: R→ R $ is continuous at $ a $, then $ g : x→ |f(x)| $ is also continuous at $a $ . I don't know how to approach this, how is it proved?

Answer 1

Hint:

By the triangular inequality,

$$||f(x)|-|f(a)||\le|f(x)-f(a)|.$$

Answer 2

Let $f :R \rightarrow R$ be a continuous function at $a$. Then $\forall \epsilon > 0, \text{ there exists a } \delta $ s.t. $\left|x-c\right|<\delta \implies \left|f(x)-f(c)\right| < \epsilon$.

Not consider $g(x) = |f(x)|$. Let $\epsilon > 0$. We know by previous definition that there exists a $\delta$ s.t. $\left|x-c\right|<\delta$ implies $\left|f(x)-f(c)\right| < \epsilon$. By the reverse triangle inequality we have:

$$\Big||f(x)| - |f(c)|\Big| \le |f(x) - f(c)| < \epsilon$$

and therefore $\left|x-c\right|<\delta$ implies $\Big||f(x)| - |f(c)|\Big| < \epsilon$ as required.