A continuous function can be bounded from above by a smooth 'adapted' function?

Consider a continuous function $f:\mathbb{R}\to\mathbb{R}$. Is there a standard strategy that produces a function $g\in C^{\infty}(\mathbb{R},\mathbb{R})$, such that

1. $f(t)\le g(t)$$\forall t$;

2. $g(t)$ can be defined only by $\{f(s)\}_{s\le t}$.

A similar $g $ function allows to produce a smooth upper bound to the function $t\to max_{x \in [0,1]}h(x,t)$, where $h\in C^{\infty}([0,1]\times \mathbb{R},\mathbb{R}_+)$.

I highlight that the second requirement is related to probabilistic concepts such as adaptability.

Thank you in advance for any hint!