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We remember that we have already looked at the concept of continuity for ordinary functions. In this article, we will try to relate it to multivariable functions. Let us get started.
Re-visiting Continuity for Ordinary Functions
Let us first remember what is meant by continuity for ordinary functions. Intuitively, a function is continuous when you can draw its graph without lifting your pencil. This simply means no holes and no jumps in the graph. Let us take a look at two discontinuous graphs and a continuous one. Afterward, we can look at a more formal definition. We have the graphs:
In the first graph, we can see that our graph is continuous everywhere on its domain — we can draw it without lifting our hands. In the second graph, we can see that it’s jump-discontinuous. We cannot draw it with one single line — and instead, it is discontinuous in the point where the two lines are different. In the last one, we can see that we have a hole in our graph since one point is defined away from the rest of the graph. Hence, the graph is discontinuous at that point.
Let us then get a definition of continuity. It can be defined as:
From this definition, we can conclude the following fact:
