# That given a continuous surface, contour lines exist

#### JamesTheBond

Can you guys help me prove: Given a continuous and differentiable function (or surface) f: R^2 -> R, such that f(x,y) = z ... contour lines can always be drawn... the function is NOT bijective.

I've been thinking of choosing any arbitrary point and showing that the curves that intersect to form that point (over y-axis and x-axis) always manage to have certain z-values that are equal.

How do I point this is in mathematical terms?

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#### JamesTheBond

Here is a start of my solution which I am stuck with.

Consider this continuous surface $$f(x,y) = z$$. Consider arbitrary surface point $$(i,j,z _i_j)$$. This point must exist (given $$f(i,j) = z _i_j$$).

Consider continuous functions $$f(x,j)$$ and $$f(i,y)$$ which are curves that are part of the surface. $$f(x,j) = f(i,y) = z_i_j$$ are equal IFF x = i and y = j.

If I can disprove the IFF part, then I should be done right?

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#### Hurkyl

Staff Emeritus
Gold Member
I've been thinking of choosing any arbitrary point
Let P = (a, b, f(a, b)) be a point on the surface.

and showing that the curves that intersect to form that point (over y-axis and x-axis)
Consider the curves C = (x, b, f(x, b)) and D = (a, y, f(a, y)) that lie on the surface.

always manage to have certain z-values that are equal.
There exists an s and a t such that (s, b, f(s, b)) and (a, t, f(a, t)) have the same z-coordinate.

Incidentally, this hypothesis is false. For example, consider the point (0, 0, 0) on the surface:
z = x^2 - y^2​
The curve over the x-axis never shares a z-coordinate with a point over the y-axis. (Except at the origin when both z-coordinates are zero)

Hint: it might help to think, not of a surface in R^3, but of a function R^2-->R. (And I think you have one of the right ideas for proving this)

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#### JamesTheBond

Hmm.... I can't just use the curves along x and y axes. Because there are bound to be such R^2 -> R functions which elude only those particular axes?

#### JamesTheBond

The function is everywhere differentiable and continuous/

#### AlephZero

Homework Helper
The function is everywhere differentiable and continuous/
Therefore at any point you can define a tangent plane. If the tangent plane is not horizontal then you can define the direction of the contour line using it.

If the tangent plane is horizontal then you need to refine what you mean by "drawing contours". There may be a finite number (including zero) or an infinite number of "contour lines" through such a point.

Examples: functions f(x,y) = x^2 + y^2. f(x,y) = x^2 - y^2, f(x,y) = 0 at the origin.

Staff Emeritus