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- https://math.libretexts.org/Courses/Montana_State_University/M273%3A_Multivariable_Calculus/14%3A_Functions_of_Multiple_Variables_and_Partial_Derivatives/Limits_and_ContinuityA function of two variables is continuous at a point (x0,y0) in its domain if for every ε>0 there exists a δ>0 such that, whenever \sqrt{(x−x_0)^2+(y−y_0)^2}<δ it is true, \(|f(x,y)−...A function of two variables is continuous at a point (x_0,y_0) in its domain if for every ε>0 there exists a δ>0 such that, whenever \sqrt{(x−x_0)^2+(y−y_0)^2}<δ it is true, |f(x,y)−f(a,b)|<ε. This definition can be combined with the formal definition (that is, the epsilon–delta definition) of continuity of a function of one variable to prove the following theorems:
- https://math.libretexts.org/Courses/Misericordia_University/MTH_226%3A_Calculus_III/Chapter_14%3A_Functions_of_Multiple_Variables_and_Partial_Derivatives/3.04%3A_Limits_and_ContinuityA function of two variables is continuous at a point (x_0,y_0) in its domain if for every ε>0 there exists a δ>0 such that, whenever \sqrt{(x−x_0)^2+(y−y_0)^2}<δ it is true, \(|f(x,y)−...A function of two variables is continuous at a point (x_0,y_0) in its domain if for every ε>0 there exists a δ>0 such that, whenever \sqrt{(x−x_0)^2+(y−y_0)^2}<δ it is true, |f(x,y)−f(a,b)|<ε. This definition can be combined with the formal definition (that is, the epsilon–delta definition) of continuity of a function of one variable to prove the following theorems:
- https://math.libretexts.org/Courses/Georgia_State_University_-_Perimeter_College/MATH_2215%3A_Calculus_III/14%3A_Functions_of_Multiple_Variables_and_Partial_Derivatives/Limits_and_ContinuityA function of two variables is continuous at a point (x_0,y_0) in its domain if for every ε>0 there exists a δ>0 such that, whenever \sqrt{(x−x_0)^2+(y−y_0)^2}<δ it is true, \(|f(x,y)−...A function of two variables is continuous at a point (x_0,y_0) in its domain if for every ε>0 there exists a δ>0 such that, whenever \sqrt{(x−x_0)^2+(y−y_0)^2}<δ it is true, |f(x,y)−f(a,b)|<ε. This definition can be combined with the formal definition (that is, the epsilon–delta definition) of continuity of a function of one variable to prove the following theorems:
