Can Matlab plot a 1st-order ODE that has initial conditions at two values of t? f'(t) = f(2t)

I'm trying to plot the function shown above (generated in Excel), defined as a first-order Ordinary Differential Equation (ODE), which looks deceptively simple, but has proven difficult to analyze. It is simply:

f'(t) = f(2t), where
f(0) = 0

Thus,

f'(0) = f(0) = 0

As defined above, there exists a trivial solution: f(t) = 0. And for a "sufficiently smooth" function, there is exactly one solution. Apparently, I am not looking at a smooth function. I want a solution that also meets the following conditions:

f(2) = 1
f'(2) = 0

It follows that

f'(1) = f(2) = 1

Due to the function's symmetry,

f(1) = 0.5

It follows from this that

f'(0.5) = f(1) = 0.5

In studying Matlab's ODE solvers, it appears that it calculates each new iteration N from the value of iteration N-1 (or in the case of ode113, from multiple previous iterations). Thus, I don't see how the solver can take into account the values at t=2.

There is a theorem (I forget the name) that states that if all the derivatives of a function are known at some time t0, and all the function's derivatives are finite over an interval that contains time t0, then the function is completely defined over that interval. In the case of this function, f'(t) = f(2t), and f(0) = 0. So 0 = f'(0) = f''(0) = f'''(0) and so on: all higher derivatives at t=0 are zero, and the function is simply f(t) = 0. However, this function as I am defining it does not satisfy the conditions of the above theorem because the derivatives at times other than t=0 approach infinity as the order of the derivatives approaches infinity. Nevertheless, the function is continuous in all its derivatives and has a very smooth, simple and elegant shape. The reason why I want to plot the function in Matlab is to verify that the function I cumbersomely generated in Excel is indeed the function f'(t) = f(2t).

NOTE: This function appears to be periodic, but is not. Also, if the function can be defined over the interval 0 <= t <= 1, it can be easily extrapolated to any value of t.