If $f(x,t) : \mathbb{T}^3 \times (0,\infty) \to \mathbb{R}$ solves the periodic heat equation and $f(x,0)=0$, is it true that $f$ is identically zero?

Question

Let $\mathbb{T}^3 : = (\mathbb{R}/\mathbb{Z})^3$ be the $3$-dimensional torus and $f(x,t) : \mathbb{T}^3 \times [0,\infty) \to \mathbb{R}$ be a function such that

1. $f$ is smooth on $\mathbb{T}^3 \times (0,\infty)$.

2. $f(x,0)=0$ for all $x \in \mathbb{T}^3$

3. $\partial_t f-\Delta_x f=0$ on $\mathbb{T}^3 \times (0,\infty)$.

Then, "without" a priori assumption of continuity for $f(x,t)$ at $t=0$, can we still conclude that $f(x,t)=0$ for "all" $(x,t) \in \mathbb{T}^3 \times [0,\infty)$?

I guess this is true, but cannot prove / disprove my conjecture rigorously.

Could anyone please help me?