By a distribution, I mean it is a linear functional of the space of smooth compactly supported functions over $\mathbb R^n$, i.e. $C_c^{\infty}(\mathbb R^n).$
I am reading a textbook by Strichartz, named A Guild to Distribution Theory and Fourier Transforms.
He wrote that linear functionals all tend to be continuous.
As we know, there are plenty of linear functionals which are not continuous. So what he referred to may be distributions. i.e. Distributions all tend to be continuous.
He gave a rough explanation which I think is not a proof.
His explanation:
Let $\varphi$ and $\varphi_1$ be in $C_c^{\infty}(\mathbb R^n),$ and $f$ be a distribution.
And let $\varphi_2:=\varphi_1-\varphi$
Then $\varphi_1=\varphi+\varphi_2$
Move $\varphi_1$ closer to $\varphi$ by considering $\varphi+t\,\varphi_2$ and let $t$ get small.
Then $\langle f,\varphi+t\,\varphi_2\rangle=\langle f,\varphi\rangle+t\,\langle f,\varphi_2\rangle$ by linearity, and as $t$ gets small this gets close to $\langle f,\varphi\rangle.$
End of explanation.
I think, to prove the continuity of $f$, we need to show $\langle f,\varphi_1\rangle\to\langle f,\varphi\rangle$ when $\varphi_1 \to \varphi.$
While in the explanation, what he proved is that $\langle f,\varphi+t\,\varphi_2\rangle\to\langle f,\varphi\rangle$ when $t\to 0.$
He also wrote that this does not constitute a proof of continuity, since the definition requires more "uniformly", but it should indicate that a certain amount of continuity is built into linearity.
And, all distribution you will ever encounter will be continuous.
So my problem is, are distributions all continuous?
Thanks in advance.