Prove that $f:\mathbb R\to\mathbb R^2$ is continuous where $f(x)=(x^2-5,\frac{1}{x^2+1})$

Question

Prove that $f:\mathbb R\to\mathbb R^2$ is continuous where $f(x)=(x^2-5,\frac{1}{x^2+1})$

My try:If this was from $f:\mathbb R^2\to\mathbb R$,then I may solve it.Kindly help!!!.

Answer 1

A function is continuous iff every component function is continuous. That is, you have to check for every component of $f(x)$, in your case $x^2-5$ and $\frac1{x^2+1}$, whether the functions $f_1\colon \mathbb R\rightarrow \mathbb R\colon x\mapsto x^2-5$ and $f_2\colon \mathbb R\rightarrow \mathbb R\colon x\mapsto \frac1{x^2+1}$ are continuous. If both are continuous, then so is $f$, otherwise it isn't.