Dear Colleagues,

Could you please help in solving the following problem:

Let $\displaystyle T

(T)\longrightarrow Y $ be a linear operator, where $\displaystyle D(T)\subset X$ and $\displaystyle X, Y$ are normed spaces. Then:

If $\displaystyle T$ is continuous at a single point $\displaystyle x_{0}\in X$ with $\displaystyle x_{0}\neq0$, then it is continuous at every point of $\displaystyle X$. Here $\displaystyle D(T)$ denotes the domain of $\displaystyle T$.

Remark: We want to prove this proposition without using the boundedness of the operator.

Regards,

Raed.