In your case the domain of the operator is $UCB^2(R)$ (uniformly continuous and bounded functions up to the second derivative) and $f$ is continuous with values in the domain of the operator. Semigroup theory yields that the mild solution is a classical one, that you are looking for. In the case of $D^2$, you can write down the explicit formula for the solution $u$ and check that $u_{xx}$ exists, by differentiating $f$ under the integral. Then one needs an argument for $u_t$...this is the point where I prefer semigroup theory. Another possibility is to approximate $f$ with $f_n$, better in $t$, consider $u_n$ the corresponding solutions and let $n \to \infty$. Then $u_n \to u$, $(u_{n)_{xx}} \to u_{xx}$, using the fundamental solution, and then, by difference, $(u_n)_t$ also converges.

In your case the domain of the operator is $UCB^2(R)$ (uniformly continuous and bounded functions up to the second derivative) and $f$ is continuous with values in the domain of the operator. Semigroup theory yields that the mild solution is a classical one, that one you are looking for. In the case of $D^2$, you can also write down the explicit formula for the solution $u$ and check that $u_{xx}$ exists, by differentiating $f$ under the integral. Then one needs an argument for $u_t$...this is the point where I prefer semigroup theory. Another possibility is to approximate $f$ with $f_n$, better in $t$, consider $u_n$ the corresponding solutions and let $n \to \infty$. Then $u_n \to u$, $(u_{n)_{xx}} \to u_{xx}$, using the fundamental solution, and then, by difference, $(u_n)_t$ also converges.