1. Continuous Linear Operators

Dear Colleagues,

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.

2. Originally Posted by raed
Dear Colleagues,

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 $\displaystyle T$.
What have you done as of yet? The idea is that for any $\displaystyle x\in X$ $\displaystyle \displaystyle \lim_{z\to x}T(z)=\lim_{z\to x_0}T(x_0+x-z)=T(x_0)+T(x)-\lim_{z\to x_0}T(z)=T(x)$. Formalize that.