Infinite marginal utility for vanishing consumption: Is this always true?

Question

I am considering a utility function over a single good for an intertemporal allocation problem. The class of power utility functions $$ u(x) = \frac{x^\gamma}{\gamma} $$ where $\gamma < 1$, $\gamma \neq 0$ has derivative $$ u'(x) = x^{\gamma-1} $$ and thus satisfies $$ \lim_{x \to 0} u'(x) = \infty\ . $$ This makes sense to me intuitively: if I am not consuming anything at all, the marginal utility I gain from increasing my consumption even a little bit will be unbounded.

I would like to know if this holds in general. That is, can an argument be made that marginal utility must be unbounded at zero for every (single-good) utility function? References are appreciated.

Answer 1

The reason for using a power utility function is typically not that it fits some intuition (and I also don't think this intuition is particularly convincing). It is rather that these utility functions satisfy the Inada conditions and thus equations of the form $u'(x)=c$ have a unique solution for all $c>0$. So it's an assumption made mostly for convenience and simplicity.