Image of Real Square Function
Jump to navigation
Jump to search
Theorem
The image of the real square function is the entire set of positive real numbers $\R_{\ge 0}$.
Proof
From Square of Real Number is Non-Negative, the image of $f$ is $\R_{\ge 0}$.
From Positive Real has Real Square Root:
- $\forall x \in \R: \exists y \in \R: x^2 = y$
Hence the result by definition of image.
$\blacksquare$
Sources
- 1964: William K. Smith: Limits and Continuity ... (previous) ... (next): $\S 2.2$: Functions