Setting the function at intervals

I have a finite series of intervals of real numbers, R _i= (R _imin, R _imax) and a series of real numbers t _i, i = 1..N.

My goal is to find a function f: R-> R where for each i there is f (t _i) in the interval R _i...

In the following image, the x-axis is the t values _i under each red line that correspond to the intervals R _i, and the green line is one of the possible solutions (this is a random constant).

I know that I need the function f to be continuous and differentiable to at least the third degree, and it also needs to be "as smooth as possible". When one can be linear, it should be. I was thinking of a solution where I would put the midpoints of the intervals with some spline, but that would cause over-setting problems, and it is clear that the function could be "smoother" in a way, although I don't have an exact metric for that. In my example, the image will create a clearly bad solution, and this will be the case even if there is no linear solution.

I know that this criterion for "smoothness" is somehow vague. The f function will move the car along one axis in time, so I need it to move as little as possible without any jumps or rapid changes in speed, but I don't want to define this precisely as this will narrow the approaches.

I have never encountered a problem like this, either in work or during my studies, and I don't know if it has some kind of standard name that I could Google and research further. I tried to find descriptions and keywords of my problem, but with no success.

I don't know if this is a question for SO or MO, but I need to create an algorithm to find the function f, so I am posting it here.

Any help would be much appreciated.

Matei