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Here is the full question: Let $X$ be a topological space and $f,g$ two continuous functions $f:X\to \Bbb R$. Prove that the functions $h(x)=|f(x)|$ and $k(x)=\max\{f(x),g(x)\}$ are both continuous.

I am having trouble proving the first part that $h(x)=|f(x)|$ is continuous.

Here's my answer for the second part:

We know the maximum of two functions $f(x)$ and $g(x)$ is defined as $max\{f(x),g(x)\}=\frac{f(x)+g(x)}{2}+\frac{|f(x)-g(x)|}{2}$. Since this is the sum of two continuous functions $f(x)$ and $g(x)$, then by the properties of continuous functions, $k(x)$ is also a continuous function.

Could someone point me in the right direction for the first part?

Bernard
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popcornsr
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1 Answers1

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The function $a(x)=|x|$ is continuous on $\Bbb R$. You can do an appeal to calculus, or apply the pasting lemma to

$$a(x) = \begin{cases} x & x \in [0,+\infty)\\ -x & x \in (-\infty, 0] \end{cases}$$

using that both sets are closed and $x \to x$ and $x \to -x$ are continuous on those domains. Or use the reverse triangle inequality

$$|a(x)-a(y)|= \left|\, |x| - |y|\, \right| \le |x-y|$$

to see $a$ is uniformly continuous..

Then $|f| = a \circ f$, a composition of continuous maps.

Henno Brandsma
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