Labware - MA35 Multivariable Calculus - Two Variable Calculus
 MA35 Labs 2 » Two Variable Calculus Contents2.1 Functions of Two Variables 2.1.1 Introduction 2.1.2 Linear Functions 2.1.4 Slice Curves 2.1.6 Continuity 2.4 Integration Search

Continuity (Page: 1 | 2 | 3 | 4 )

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If the coordinate functions x(t) and y(t) are continuous functions of the parameter t, then the function that sends t to the point (x(t),y(t)) is continuous.

Demos

 Continuity of Parametric Functions Define an interval for t, and two functions x(t), y(t). In a three-dimensional graph, show (t, x(t),0), (t, 0, y(t)), and (0,x(t),y(t)). For a given t0 and interval on the t-axis determined by a &delta, show the strip above this interval in the first plane and in the second plane, and show the ε/2 strip about x(t0) in the third plane and the ε/2 strip about y(t0 in the third plane, intersecting in a square region completely contained in ε disc about (x{t0),y(t0)) in the third plane. Choosing &delta small enough will make the &delta strips lie in the respective ε/2 intervals, so the image of the parametric curve will lie in the square therefore in the disc.