Finding domain for when composite function is continuous

[ChiralSuperfields]

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!

 

[fresh_42]

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?

 

[Mark44]

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.

 

[pasmith]

Note that $\ln x^2 = 2\ln |x|$.