)boneill3 said:
Continuity of a two variable function means that the function has no abrupt changes or breaks in its graph and can be drawn without lifting the pen from the paper.
Continuity refers to the smoothness of a function, while differentiability refers to the existence of the derivative at a point. A function can be continuous but not differentiable, but if a function is differentiable, it must also be continuous.
Continuity is important because it ensures that the function is well-behaved and predictable. It also allows us to make accurate calculations and predictions using the function.
A two variable function is continuous if the limit of the function at a point exists and is equal to the value of the function at that point. This can be tested using the definition of continuity or by checking if the function satisfies the three conditions of continuity: the function is defined at the point, the limit exists at the point, and the limit is equal to the value of the function at that point.
Some common examples of discontinuous two variable functions include step functions, absolute value functions, and piecewise functions. These functions have abrupt changes or breaks in their graphs, making them discontinuous.