  In this work, we establish pathwise functional It\^o formulas for non-smooth
functionals of real-valued continuous semimartingales. Under finite
$(p,q)$-variation regularity assumptions in the sense of two-dimensional Young
integration theory, we establish a pathwise local-time decomposition
  $$F_t(X_t) = F_0(X_0)+ \int_0^t\nabla^hF_s(X_s)ds +
\int_0^t\nabla^wF_s(X_s)dX(s) -
\frac{1}{2}\int_{-\infty}^{+\infty}\int_0^t(\nabla^w_xF_s)(^{x}X_s)d_{(s,x)}\ell^x(s).$$
Here, $X_t= \{X(s); 0\le s\le t\}$ is the continuous semimartingale path up to
time $t\in [0,T]$, $\nabla^h$ is the horizontal derivative, $(\nabla^w_x
F_s)(^{x}X_s)$ is a weak derivative of $F$ with respect to the terminal value
$x$ of the modified path $^{x}X_s$ and $\nabla^w F_s(X_s) = (\nabla^w_x
F_s)(^{x} X_s)|_{x=X(s)}$. The double integral is interpreted as a space-time
2D-Young integral with differential $d_{(s,x)}\ell^x(s)$, where $\ell$ is the
local-time of $X$. Under less restrictive joint variation assumptions on
$(\nabla^w_x F_t)(^{x} X_t)$, functional It\^o formulas are established when
$X$ is a stable symmetric process. Singular cases when $x\mapsto (\nabla^w_x
F_t)(^{x}X_t)$ is smooth off random bounded variation curves are also
discussed. The results of this paper extend previous change of variable
formulas in Cont and Fourni\'e and also Peskir, Feng and Zhao and Elworhty,
Truman and Zhao in the context of path-dependent functionals. In particular, we
provide a pathwise path-dependent version of the classical
F\"{o}llmer-Protter-Shiryaev formula for continuous semimartingales.
