# Show that max(f,g) is continious?is my answer wrong?

## Homework Statement

let f and g are continuous functions show that max(f,g) is continious

## The Attempt at a Solution

the answer i gave is this
maxf(g)=h(x)
h(x)=f(x)==>f(x)-g(x)>0
h(x)=g(x)==>f(x)-g(x)≤0
thus h is continious

now in the text book they defined max with the normal definition
max=lf(x)-g(x)l+f(x)+g(x)/2

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## Homework Statement

let f and g are continious functions show that max(f,g) is continious

## The Attempt at a Solution

the answer i gave is this
maxf(g)=h(x)
h(x)=f(x)==>f(x)-g(x)>0
h(x)=g(x)==>f(x)-g(x)≤0
thus h is continious
Why does this imply that h is continuous?

I thought it's obvious
if f(x)>g(x) then h(x) = f(x) f continious then h continious
if f(x)<=g(x) then h(x)=g(x) g gotinious thus h is continious
Am i doing something wrong here?(spivak's calculus questions are quiet hard )
Edit : i wasn't supposed to study cases?

Mark44
Mentor

## Homework Statement

let f and g are continious functions show that max(f,g) is continuous
"Continious" is not a word - the one you mean is continuous.

## The Attempt at a Solution

the answer i gave is this
maxf(g)=h(x)
Presumably you mean max(f, g) = h(x)
h(x)=f(x)==>f(x)-g(x)>0
h(x)=g(x)==>f(x)-g(x)≤0
The above assume that f > g for all x in the domain or that g ≤ f for all x in the domain. What happens if one function is greater than the other in some places, but not in others?
thus h is continious

now in the text book they defined max with the normal definition
max=lf(x)-g(x)l+f(x)+g(x)/2
For absolute value, use |, not l (lower-case "ell"). Also, what gets divided by 2? As you wrote it, only g(x) is divided by 2. If that's not what you meant, use parentheses.

This looks like the following proof:

Consider ##f(x) = 1## for all ##x##, ##g(x) = -1## for all ##x## and take

$$h(x) = \left\{\begin{array}{l} 1~\text{if}~x<0\\ -1~\text{if}~x\geq 0\end{array}\right.$$

If ##h(x)<0##, then ##h(x) = g(x)## and ##g## is continuous.
If ##h(x)>0##, then ##h(x) = f(x)## and ##f## is continuous.
Thus ##h## is continuous.

This is essentially what you did. But the conclusion that ##h## is continuous is false of course.

The above assume that f > g for all x in the domain or that g ≤ f for all x in the domain. What happens if one function is greater than the other in some places, but not in others?
i was missing this part thank you
micromass made me laugh a bit at my "proof" :).
thanks guys .