Why does Tao call it *Continuity of exponentiation*?

Question

I’m self-studying Tao’s Analysis I. He states the following.

Lemma 6.7.1 (Continuity of exponentiation). Let $x > 0$, and let $α$ be a real number. Let $(q_n)_{n=1}^\infty$ be any sequence of rational numbers converging to $α$. Then $(x^{q_n})_{n=1}^\infty$ is also a convergent sequence. Furthermore, if $(q’_n)_{n=1}^\infty$ is any other sequence of rational numbers converging to $α$, then $(x^{q’_n})_{n=1}^\infty$ has the same limit as $(x^{q_n})_{n=1}^\infty$:

$\lim\limits_{n\to\infty} x^{q_n}=\lim\limits_{n\to\infty} x^{q’_n}.$

Question: Why does he call this Continuity of exponentiation? What is continuity in this?

Answer 1

He is just saying that the map $f(q):=x^q$ is continuous.