Limits and Continuity: Learn It 2

Interior Points and Boundary Points

To study continuity and differentiability of a function of two or more variables, we first need to learn some new terminology.

interior and boundary points

Let [latex]S[/latex] be a subset of [latex]\mathbb{R}^{2}[/latex].

  • A point [latex]P_0[/latex] is an interior point of [latex]S[/latex] if there is a [latex]\delta[/latex] disk centered around [latex]P_0[/latex] contained completely in [latex]S[/latex].
  • A point [latex]P_0[/latex] is a boundary point of [latex]S[/latex] if every [latex]\delta[/latex] disk centered around [latex]P_0[/latex] contains points both inside and outside [latex]S[/latex]
On the xy plane, a closed shape is drawn. There is a point (–1, 1) drawn on the inside of the shape, and there is a point (2, 3) drawn on the boundary. Both of these points are the centers of small circles.
Figure 4. In the set [latex]S[/latex] shown, [latex]\small{(-1,1)}[/latex] is an interior point and [latex]\small{(2,3)}[/latex] is a boundary point.

open and closed sets

Let [latex]S[/latex] be a subset of [latex]\mathbb{R}^{2}[/latex].

  • [latex]S[/latex] is an open set if every point of [latex]S[/latex] is an interior point.
  • [latex]S[/latex] is a closed set if it contains all its boundary points.

An example of an open set is a [latex]\delta[/latex] disk. If we include the boundary of the disk, then it becomes a closed set. A set that contains some, but not all, of its boundary points is neither open nor closed. For example, if we include half the boundary of a [latex]\delta[/latex] disk but not the other half, then the set is neither open nor closed.

connected sets and regions

Let [latex]S[/latex] be a subset of [latex]\mathbb{R}^{2}[/latex].

  • An open set [latex]S[/latex] is connected if it cannot be represented as the union of two or more disjoint, nonempty open subsets.
  • A set [latex]S[/latex] is a region if it is open, connected, and nonempty.

Limits at Boundary Points

The definition of a limit of a function of two variables requires the [latex]\delta[/latex] disk to be contained inside the domain of the function. However, if we wish to find the limit of a function at a boundary point of the domain, the [latex]\delta[/latex] disk is not contained inside the domain. By definition, some points of the [latex]\delta[/latex] disk are inside the domain and some are outside.

Therefore, we need only consider points that are inside both the [latex]\delta[/latex] disk and the domain of the function. This leads to the definition of the limit of a function at a boundary point.

limit at a boundary point

Let [latex]f[/latex] be a function of two variables, and suppose [latex](a,\ b)[/latex] is on the boundary of the domain of [latex]f[/latex]. Then

[latex]\underset{(x,y)\to(a,b)}{\text{lim}}f(x,y)=L[/latex]

if for any [latex]\epsilon>0[/latex], there exists [latex]\delta>0[/latex] such that

[latex]|f(x,y)-L|<\epsilon[/latex] whenever [latex]0<\sqrt{(x-a)^{2}+(y-b)^{2}}<\delta[/latex]

for all points [latex](x,\ y)[/latex] inside the domain of [latex]f[/latex].

Prove [latex]\displaystyle\lim_{(x,\ y)\to(4,\ 3)}\sqrt{25-x^{2}-y^{2}}=0[/latex].