Let $K(x,t):=\frac{1}{(4\pi t)^{n/2}}e^{-\lvert x \rvert^2/(4t)}$ be the $n$-dimensional heat kernel.
Also, consider a locally-integrabl functon $f : \mathbb{R}^n \times [0,\infty) \to \mathbb{R}$. That is, for any compact subset $K \subset \mathbb{R}^n$ and $t \in (0,\infty)$, we have \begin{equation} \int_0^t \int_K \lvert f(x,\tau) \rvert dx d\tau < \infty. \end{equation}
Now, let $B_1 \subset \mathbb{R}^n$ be the closed unit ball centered at the origin. For each $(x,t) \in B_1 \times (0,\infty)$, define \begin{equation} (K*f)(x,t):=\int_{t/2}^t \int_{B_1} K(y,\tau) f(x-y,t-\tau)dyd\tau. \end{equation}
Then, I can see that it is a well-defined continuous function on $B_1 \times (0,\infty)$. However, I wonder if this function "extends continuously to $B_1 \times [0,\infty)$ as well.
That is, if $x_m \to x$ in $B_1$ and $t_m \to 0^+$, do we still have \begin{equation} (K*f)(x_m,t_m) \to 0 \text{ as } m \to \infty \end{equation} ?
I guess (and hope) that this is right, due to the definition of $K * f$. However, it is quite subtle to prove / disprove it rigorously since $f$ is NOT assumed to be continuous... Could anyone please help me?