g-1(X) will have jumps at g-1(x1), g-1(x2), g-1(x3)

So if g is left continuous, g-(1) will be right continuous and vice versa.

An easy way to see is that the graph of inverse of a function is just the axes flipped(interchanged)

So if you turn your notebook 90 degree, look at the y axis as new x, and x axiis as new y, what you'll see is the graph of g-1(x), which will be right continuous if g is left continuous.