Is it possible to compute the wavelet transform of a unit step function?
Finite energy functions is a class of choice, but wavelet transforms can exists in more generic cases. 

For the continuous case: let us call $H(t)$ the Heaviside step function. Then $|H(t)\psi(t)|$ is dominated by $|\psi(t)|$. If $\psi(t)$ is $L_1$ or integrable (in addition to being  $L_2$), then  $\int_{-\infty}^{\infty}H(t)\psi(\frac{t-a}{b})\mathrm{d}t$ is defined. Moreover, since admissible wavelets have zero-average, the continuous wavelet coefficients can decay at infinity. With wavelet of finite support, they will be zero for all $\psi(\frac{t-a}{b})$ with support in $[0,\;+\infty[$.

For discrete wavelets, this would be about the same. As long as the filters are summable (or $\ell_1$), and this happens  especially for FIR wavelet filters, you can compute coefficients. Special note: if a discrete filter is $\ell_1$, then it is $\ell_2$ as well, so the condition is milder than in the continuous case.