Finding domain for when composite function is continuous

member 731016
Homework Statement
Please see below
Relevant Equations
Please see below
I am trying to find where ##h(x) =In{x^2}## is continuous on it's entire domain.

My reasoning is since natural log is defined for ##x > 0##, then the argument ##x^2## should be positive, ##x^2 > 0##, we can see without solving this equation that ##x ≠ 0## for this equation to be true, however, does someone please know how we could prove this by solving that equation for x?

My working is
##x > 0## (Taking square root of both sides of the equation)

Many thanks!
 
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ChiralSuperfields said:
Homework Statement: Please see below
Relevant Equations: Please see below

I am trying to find where ##h(x) =In{x^2}## is continuous on it's entire domain.

My reasoning is since natural log is defined for ##x > 0##, then the argument ##x^2## should be positive, ##x^2 > 0##, we can see without solving this equation that ##x ≠ 0## for this equation to be true, however, does someone please know how we could prove this by solving that equation for x?

My working is
##x > 0## (Taking square root of both sides of the equation)

Many thanks!
It is neither defined nor continuous at ##x=0.## It is continuous everywhere else. What do you use to prove continuity? E.g. it is differentiable at ##x\neq 0## and therewith continuous. Or you use a definition for continuity. There are a few, so which one do you use?
 
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ChiralSuperfields said:
I am trying to find where ##h(x) =In{x^2}## is continuous on it's entire domain.
There is no "##In()## function; i.e., starting with uppercase i. It's ##\ln()##, with a lowercase letter l (ell), short for logarithmus naturalis.
 
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Note that \ln x^2 = 2\ln |x|.
 
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Prove $$\int\limits_0^{\sqrt2/4}\frac{1}{\sqrt{x-x^2}}\arcsin\sqrt{\frac{(x-1)\left(x-1+x\sqrt{9-16x}\right)}{1-2x}} \, \mathrm dx = \frac{\pi^2}{8}.$$ Let $$I = \int\limits_0^{\sqrt 2 / 4}\frac{1}{\sqrt{x-x^2}}\arcsin\sqrt{\frac{(x-1)\left(x-1+x\sqrt{9-16x}\right)}{1-2x}} \, \mathrm dx. \tag{1}$$ The representation integral of ##\arcsin## is $$\arcsin u = \int\limits_{0}^{1} \frac{\mathrm dt}{\sqrt{1-t^2}}, \qquad 0 \leqslant u \leqslant 1.$$ Plugging identity above into ##(1)## with ##u...

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