Why does a maximum flow in a transportation network always exist In a transportation network the Ford Fulkerson algorithm and others are often used to determine the maximum flow. How do we know that there is such a thing as a flow which is a maximum flow in the first place?
Van Lint's book "A course in combinatorics" which I am reading from says that the maximum flow exists by "continuity reasons". I do not understand this.
 A: You have some assumptions to work with here that guarantee a maximum flow (the flow is the amount, there may be many distributions of that flow across the graph that work). For one, it is typical in graph theory to assume the graph is finite unless otherwise stated (this is common but not universal, so, check with your author's conventions; hopefully they pointed that out). Additional, since you're talking about a flow network, that means you have 1 source and 1 sink. The specific reason he refers to continuity reasons is that you can represent the space of possible flows (those that abide by the constraints of the problem, maximum or not), as a convex subset of $(\mathbb{R}^+)^{|E|}$, where $|E|$ denotes the number of edges in your graph and is the 'exponent' on $\mathbb{R}^+$. Because that set is a convex subset of $\mathbb{R}^{|E|}$, you use analysis theorems to show that there is a point on it's boundary of maximal 'distance' (distance being measured by the 'taxicab' metric, which is the sum of absolute values of the changes in each variable, not euclidean distance) from the origin (maximum flow is bounded by the constraints on edges and the finiteness of the graph).
