  Avikainen provided a sharp upper bound of the difference
$\mathbb{E}[|g(X)-g(\widehat{X})|^{q}]$ by the moments of $|X-\widehat{X}|$ for
any one-dimensional random variables $X$ with bounded density and
$\widehat{X}$, and function of bounded variation $g$. In this article, we
generalize this estimate to any one-dimensional random variable $X$ with
H\"older continuous distribution function. As applications, we provide the rate
of convergence for numerical schemes for solutions of one-dimensional
stochastic differential equations (SDEs) driven by Brownian motion and
symmetric $\alpha$-stable with $\alpha \in (1,2)$, fractional Brownian motion
with drift and Hurst parameter $H \in (0,1/2)$, and stochastic heat equations
(SHEs) with Dirichlet boundary conditions driven by space--time white noise,
with irregular coefficients. We also consider a numerical scheme for maximum
and integral type functionals of SDEs driven by Brownian motion with irregular
coefficients and payoffs which are related to multilevel Monte Carlo method.
