Let $X$ and $Y$ be topological vector spaces and $T:X\to Y$ a linear operator. If $T$ is continuous at $0\in X$ then $T$ is continuous.
Suppose $x_0\in X$ and let $U$ be a nbhd of $T(x_0)$ in $Y$. To use the assumption, I want to somehow related $U$ to a nbhd of $T(0)$. How can I use the linearity $T(x-x_0)=Tx-Tx_0$ here? I guess $U-Tx_0$ might work but I don't see why it could be a nbhd of $T(0)$ by definition.
As $Y$ is a topolgical vector space, addition is continuous and therefore the map $f \colon Y \to Y$, $y \mapsto y - Tx_0$ and its inverse, $f^{-1}\colon y \mapsto y + Tx_0$ are continuous. Hence, $f$ is a homeomorphism, and $f[U]$ is open as image of an open set under a homeomorphism.