# Math Help - Continuous Linear Operators

1. ## Continuous Linear Operators

Dear Colleagues,

Let $T(T)\longrightarrow Y " alt="T(T)\longrightarrow Y " /> be a linear operator, where $D(T)\subset X$ and $X, Y$ are normed spaces. Then:
If $T$ is continuous at a single point $x_{0}\in X$ with $x_{0}\neq0$, then it is continuous at every point of $X$. Here $D(T)$ denotes the domain of $T$.

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

Regards,

Raed.

2. Originally Posted by raed
Dear Colleagues,

Let $T(T)\longrightarrow Y " alt="T(T)\longrightarrow Y " /> be a linear operator, where $D(T)\subset X$ and $X, Y$ are normed spaces. Then:
If $T$ is continuous at a single point $x_{0}\in X$ with $x_{0}\neq0$, then it is continuous at every point of $X$. Here $D(T)$ denotes the domain of $T$.
What have you done as of yet? The idea is that for any $x\in X$ $\displaystyle \lim_{z\to x}T(z)=\lim_{z\to x_0}T(x_0+x-z)=T(x_0)+T(x)-\lim_{z\to x_0}T(z)=T(x)$. Formalize that.