Originally Posted by

**southprkfan1** Let C be the space of continuous functions from [a,b] to R (real numbers) with the metric:

d(f,g) = sup lf-gl

Is the interior of C a non-empty set?

I say it is an empty set, but I'm not sure about some of my steps. Let me know what you all think:

**Outline of Proof**

Suppose the interior is not empty and contains f.

Then, for some e>0, the open ball around f of radius e is contained in C.

But, because f is continuous, then if we break [a,b] up into small enough intervals such that on each interval [Xi, X(i+1)], there is a number Ci whereby:

If x is in [Xi, X(i+1)) -->lf(x) - Cil < e

Define a function g as:

g(x) = Ci if x is in [Xi, X(i+1))

then d(f,g) < e

so g is in the ball of radius e around f

so g is in C, but g is not continuous...contradiction.

...thoughts?