Continuous Real Function/Examples/Root of x at 1

Examples of Continuous Real Functions

Let $f: \R_{\ge 0} \to \R$ be the real function defined as:

$\map f x = \sqrt x$

Then $\map f x$ is continuous at $x = 1$.

Proof

From Limit of Real Function: Example: $\sqrt x$ at $1$, we have that:

$\ds \lim_{x \mathop \to 1} \sqrt x = 1$

The result follows by definition of continuous real function.

$\blacksquare$

Sources

• 1961: David V. Widder: Advanced Calculus (2nd ed.) ... (previous) ... (next): $1$ Partial Differentiation: $\S 2$. Functions of One Variable: $2.1$ Limits and Continuity