Prove that $\varphi\left(T\right)=T^{-1}$ are continuous where $T$ operator defined

Question

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Problem :

Let $\varphi$ application such that :

$$\varphi~~~~:~\mathcal{L}_{r}\left(E\right)\to~~~~\mathcal{L}_{r}\left(E\right)$$ $$\varphi\left(T\right)=T^{-1}$$ • prove that $\varphi$ are continuous

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$\color{red}{note}~"\color{pink}{operator~~regular}":~$

$(E,\|.\|)$ banach space & $~~T\in \mathcal{L}\left(E\right)$

$$T\in\mathcal{L}_{r}\left(E\right)\iff~T\left(E\right)=E~\wedge~T^{-1}~\text{existe}~\wedge~T^{-1}\in \mathcal{L}\left(E\right)$$

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My attempts :

We have :

$~T^{-1}\in \mathcal{L}\left(E\right)\implies \exists~C>0~:~\|T^{-1}x\|_{E}\leq~C\|x\|_{E}$

And also

$\implies~\exists~k>0~~:~~\|Tx\|\geq~k\|x\|$

So ,

$\|\varphi\|_{\mathcal{L}_{r}\left(E\right)}=\|T^{-1}\|_{\mathcal{L}_{r}\left(E\right)}$

$~~~~~~~~~~~~~~\leq~C\|x\|\leq_{k=C}$

$~~~~~~~~~~~~~~\leq\|Tx\|\leq\|T\|\|x\|$

Is my attempt mathematically correct?

Is there any other way I would to see ?

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Thanks!