Algebraic structure of continuous, unbounded and integrable functions

In this paper we study the large linear and algebraic size of the family of unbounded continuous and integrable functions in $[0,+\mathrm{\infty })$ and of the family of sequences of these functions converging to zero uniformly on compacta and in $L^1$-norm. In addition, we concentrate on the speed at which these functions grow, their smoothness and the strength of their convergence to zero.