Support Vector Classifier and cost C Applying SVC (Support Vector Classifier) to the 1-d data shown here:

What will be the support vectors for the parameter cost C=0 and C=Infinity?
As far as I read about SVM and hyper parameter C I would say:

For C = 0 there will be no support vectors


For C = Infinity all of the data points will be support vectors

Is it correct?
Note that I'm talking about SVC not SVM
SVC aims to find a separating hyperplane. That data you have is separable. Thus the hyperplane will go between x3 and x4. This are your support vectors. Now C is a parameter which allows you to make a trade off between errors allowed and the width of separating hyperplane. But your data does not need this trade off since it is already separable. Because of that the needed error to be accepted is zero. This penalty error in svc equations is multiplied with C. Because C is multiplied with 0 it does not matter the value of C, their product will be zero anyway.