Show that $f = g-h$

Question

Show that every continuous function $f$ over $\mathbb{R}$ can be written as $f = g-h$ where $g,h$ are nonnegative and continuous.

We have three cases. Either $f$ is strictly, nonnegative, nonpositive, or both. In the first case we can always break up a continuous function $f$ as $f = f-0$. Thus since $f$ and $0$ are continuous we are done. I am unsure about the other two cases, though.

Hint: You can write down $g$ and $h$ explicitly in terms of $f$ using case statements. — Thomas Andrews, Mar 09 '16 at 18:49

score 4 · Answer 1 · answered Mar 09 '16 at 18:49

4

$f^{+}:=\max\{f,0\}$ ($\max$ taken pointwise) is continuous, since

$$f^{+}=\frac{|f|+f}{2}.$$

Analogously, $f^{-}:=-\min\{f,0\}$ is also continuous. Note that $f=f^{+}-f^{-}$.

answered Mar 09 '16 at 18:49

Aloizio Macedo

34,292

How do you know $f^+$ is continuous? – Puzzled417 Mar 09 '16 at 18:58
Because of the equality after "since". $f^{+}$ is the sum of two continuous functions, divided by a constant. – Aloizio Macedo Mar 09 '16 at 18:59
How do you know that $|f|$ is continuous? – Puzzled417 Mar 09 '16 at 19:00
Do you know that the module function is continuous? If so, $|f|$ is the composition of the module function with $f$. Proving that the module function is continuous (if you don't know it) should be an easy exercise. – Aloizio Macedo Mar 09 '16 at 19:01
Basically are you saying since $f$ is continuous, so is $|f|$? – Puzzled417 Mar 09 '16 at 19:04
That's correct. Note that the converse doesn't necessarily hold. – Workaholic Mar 09 '16 at 19:04
Hmm, can someone give me a proof of that? – Puzzled417 Mar 09 '16 at 19:05
@Puzzled417 It suffices to know two facts:
1- Composition of continuous functions is continuous; 2- The map $g$ which makes $x \mapsto |x|$ (that is, the "module" function) is continuous. Because then, $|f|=g \circ f$.
– Aloizio Macedo Mar 09 '16 at 19:06
@Puzzled417 See this question. – Workaholic Mar 09 '16 at 19:08
$\vert \vert f(x) \vert-\vert f(x_0) \vert \vert \le \vert f(x)- f(x_0) \vert $ – mathcounterexamples.net Mar 09 '16 at 21:16

Show that $f = g-h$

1 Answers1