Consider the IVP, $$u_t -2txu_x = 0,\qquad u(x,0) = f(x)$$
1) suppose $f \in C^1(R)$ show that $u(x,t) = f(x\exp(t^2))$ is a strict solution of the PDE, derive the solution as well
2) Derive the weak formulation of the PDE. Assume $f(x)$ is merely piecewise continuous, show that $u(x,t) = f(x\exp(t^2))$ is a weak solution
Here is my attempt, but I do not whether I proved this is a strong solution of this IVP and what $f \in C^1(R)$ implies. Also, I do not have any clue about the second part of this problem. Any help would be appreciated! 
Yes, your first part is correct. In order for a solution to be considered "strong", it must have "just as many" derivatives as those appearing in the equation. Here, we have one time and one spacial derivative, so if f is C^1, then the proposed solution is certainly differentiable in space and time. Additionally, the proposed solution must satisfy the equation point wise almost everywhere, which is exactly what your plug in step verifies.
In order to show something is a weak solution, you must integrate both sides of the equation against an arbitrary test function and show that the equality holds for your chosen f. Hint: Integration by parts