I asked this question on math.stackexchange but deleted it from there and brought it here.
I had a question about Constant Elasticity of substitution type preferences of the form: $$U=\int_{0}^{1}(c(\omega)^{\rho}d\omega)^{\frac{1}{\rho}}$$ where the paramter $\rho$ governs the degree of substitutability between goods. Here, $c(\omega)$ represents consumption of good $\omega$ which exists on the unit interval. As such, this type of preference specification aggregates over consumption of different goods.
My question is as follows. I have always thought of integrals of the form: $$I=\int f(x)dx$$ as approximating sums of areas of infinitesimally small rectangles (in terms of their base) and heights being determined by $f(x)$ . In the case of the example above, what really is $c(\omega)$ ? Is it a function?