I know that one may find continuous $f$ (say in a ball $\bar B$ centered at $0$) such that there exists no classical ($\mathcal C^2$) solution $u$ to the Poisson equation $\Delta u =f$ in $B$, and obviously this requires to be in dimension 2 or more. (But a classical solution exists as soon as $f$ is assumed Hölder continuous). This counterexample may be found for instance in Qi Han's book "A Basic Course in Partial Differential Equations" (link) pp.136-137.
Now it is easy to see that for such an $f$ there exists no classical solution to the inhomogeneous heat equation $u_t - \Delta u = - f(x) \;\;\;\mbox{ on } (0,T) \times B.$
But the case of one space dimension seems to require a new argument.
So my question is this :
Is there a continuous function $f$ on $[0,T] \times [-a,a] $ s.t. the equation $u_t - u_{xx}=f(t,x)$ has no classical solution ?
(I know that if $f$ is Hölder continuous in $x$, uniformly in $t$, then one may prove by explicit formulas that there is a classical solution.)