Question

Determine the set of points at which the function is continuous.
\(f\left( {x,y,z} \right) = \sqrt {y{\rm{ - }}{x^2}} {l_n}z\)

Short answer

\(\left\{ {\left( {x,y,z} \right)\left| {y \ge {x^2},z{\rm{ > }}0} \right|} \right\}\)

Step-by-step solution

Step-1: Finding Restriction on Domain

The function f is a composition of elementary function and is therefore continuous on its domain.

The square root function is defined so long as the expression inside the radical is non-negative.

Therefore, one restriction on the domain is \(y{\rm{ - }}{x^2} \ge 0\)or simply \(y \ge {x^2}\).

Step-2: Finding Another Restriction

Another restriction is z>o since the natural log function is only defined for positive real numbers.

The another restriction on the domain of f and hence, the set of aa points for which f is continuous is the combination of these two restrictions.

Hence, \(\left\{ {\left( {x,y,z} \right)\left| {y \ge {x^2},z{\rm{ > }}0} \right|} \right\}\)