# Continuity of Magnetic Flux

I was trying to solve a problem in which I had to obtain the function of the magnetic flux through a loop over time. The way the magnetic field and the surface are defined is piece-wise, and I wondered whether the magnetic flux function I wanted to get had to be continuous, since it is defined as an integral (and therefore Fundamental Theorem of Calculus would imply magnetic flux's continuity):

$$\mathbf { \Phi(t) = \int_S B(t)dS}$$ The Magnetic Flux $$\Phi$$ is the Sum ( Average) of the $$\mathbf{B}$$- field over the area $$\mathbf{S}$$

If flux were to be continuous, then checking the continuity of the function I got as a result would be a nice way to know if I've solved the problem correctly or not. That's why I'm asking.

In summary, must magnetic flux over time always be a continuous function? Thanks in advance.