If $f(x)$ is discontinuous, then $u(t,x)= \mathbb{E}_xf(X(t))$ may not satisfy the initial condition, in the sense that the limit statement: $$ \lim_{(t,s) \to (0^+,x)} u(t,s) = f(x) \quad \forall x \in \mathbb{R} \tag{$\star$} $$ may not hold. To be concrete, consider $$ d X(t) = d B(t) \;, \quad X(0) = x $$ where $B(t)$ is a standard Brownian motion on $\mathbb{R}$ and let $$ f(x) = \begin{cases} 1 & \text{if $x\ge 0$} \\ 0 & \text{otherwise} \end{cases} $$ In this case, for any $t>0$: $$ u(t,x) = \mathbb{E} f(x+B(t)) = \frac{1}{2} ( 1 + \operatorname{erf}( \frac{x}{\sqrt{2t}} ) ) \;. $$ However, $ \lim_{(t,s) \to (0^+,0)} u(t,s) $ does not exist, since for any $\alpha>0$ we have that: $$ \lim_{s \to 0} u(s^2,s^{\alpha}) = \lim_{s \to 0} \frac{1}{2} ( 1 + \operatorname{erf}( \frac{s^{\alpha-1}}{\sqrt{2}} ) ) = \begin{cases} 1/2 & \alpha>1 \\ 1/2 (1 + \operatorname{erf}(1/\sqrt{2}) ) & \alpha=1 \\ 1 & \alpha < 1 \end{cases} $$ which depends on $\alpha$. Nevertheless, $u(t,x) \in C^{\infty}((0, \infty)\times \mathbb{R})$, and by dominated convergence, one can show that $u(t,x)$ satisfies the heat equation for any $t>0$. Here is a graphical illustration of the solution.
To summarize, continuity of $f(x)$ is applied to show that the limit statement $(\star)$ holds, which is why theorems relating SDEs to parabolic equations often assume that the initial condition $f$ is at least continuous.