Sum of two independent uniform random variables

Hi,

I'm looking through an example on Sheldon Ross' book "Introduction to
Probability Models" and I've some difficulty in following one of the
example in the book (refer to Example 2.35 pg 59 of 8th Edition)

In the example, if X and Y are independent random variables both
uniformly distributed on (0,1), then calculate the probability density
of X+Y.

Given probability density function of X+Y is f X+Y(a) = integral (-
infinity to +infinity) f(a-y)g(y)dy

f(a) = g(a) = 1, if 0<a<1 , and 0 if otherwise

We obtain f X+Y(a) = integral (0 to 1) f(a-y)g(y)dy

.....the solution continues....

I've having difficulty in understanding how do we determine that f(a)
= g(a) = 1 for 0<a<1 ?
I would assume that f(x) = 1 for 0<=x<=1, and g(y) = 1 for 0<=y<=1,
since both are uniform distribution U[0,1]. I'm not sure how I could
relate that f(x), g(y) relationship with f(a) & g(a).

Thanks.

.