3
$\begingroup$

Define $f\left(x\right)=\frac{\cos x}{x}$ $ f:\mathbb{R\backslash}\left\{ 0\right\} \longrightarrow\mathbb{R}$

So I need to determine if there is a continuous extension to $f\left(x\right)$ at a=0 and then find it if so.

What I thought of is since ${\displaystyle \lim_{x\to a^{-}}f\left(x\right)=-\infty}$ and $ {\displaystyle \lim_{x\to a^{+}}f\left(x\right)=\infty}$ , then it's not possible that there is a continuous extension to $f\left(x\right)$

Is that argument valid?

  • 0
    @emiliocba: I'm not sure what you mean. Viewed as a function of a complex variable, $f$ has a pole at zero, not an essential singularity. (Remember that on the Riemann sphere there is only one point at infinity.)2012-01-23

1 Answers 1

1

Yes, your argument is valid.

In fact, even half of your argument is valid: $\lim_{x \rightarrow a^+} f(x) = \infty$ is already enough to see that $f$ cannot extend to a continuous function $f: \mathbb{R} \rightarrow \mathbb{R}$.

  • 0
    @sony jimbo: If $f$ could be defined at zero so as to be continuous, then $\lim_{x \rightarrow 0} f(x)$ must exist. But since $\lim_{x \rightarrow 0^+} f(x) = \infty$, this limit does not exist.2012-01-23