Is the Set M Closed in the Space X?

[iris_m]

Homework Statement

Let X=(C([0,1]), || . ||_1 ), where ||f||_1=\int_{0}^{1}|f(t)|dt.
Let M=\{f \in C([0,1]) : \int_{0}^{1}f(t)dt=2, f(1)=0\}.
Is M closed in X?

The Attempt at a Solution

I've tried the following:
Let f_n be a sequence in M such that f_n \rightarrow f.
I'm checking whether f \in M, because that would prove that M is closed (if a set contains all the limits of its convergent sequences, it is closed).
There are obviously two conditions to check.
The first one:
\int_{0}^{1}f(t)dt=\int_{0}^{1}limf_n(t)dt=lim\int_{0}^{1}f_n(t)dt=lim 2=2.
Now I have to check the second one, that is, is f(1)=0, and I don't know how.

Any help is much appreciated.

 

[HallsofIvy]

If f_n(1)= 0 and f_n\rightarrow f, then... (What kind of convergence are you talking about? Pointwise? Uniform? In the norm?)

And is your justification for saying
\int_0^1 limf_n(t)dt= lim\int_0^1 f_n(t)dt?

 

[iris_m]

If f_n(1)= 0 and f_n\rightarrow f, then... (What kind of convergence are you talking about? Pointwise? Uniform? In the norm?)

In the norm.

And is your justification for saying
\int_0^1 limf_n(t)dt= lim\int_0^1 f_n(t)dt?

My justification would be that f_n are continuous functions, so integral and limit commute.