General question about continuity (2 var.)

Thread starter tmlfan_17
Start date Nov 15, 2011
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Continuity General

In summary, continuity in two-variable functions refers to the property of a function where small changes in the input values result in small changes in the output values. It is different from differentiability, which also requires the existence of a derivative at every point. A function can be continuous but not differentiable, which can happen when there is a sharp turn or corner in the graph. To determine continuity, we can use three conditions: the function must be defined at the point, the limit from both directions must exist and be equal, and the value of the function at the point must be equal to the limit. Continuity is important in mathematics and science because it allows for predictions and understanding of a function without evaluating every point.

Nov 15, 2011

tmlfan_17

You know that f_x(x₀,y₀) exists. What can you tell about the continuity of g(x)=f(x, y₀) at x=x₀?

I know the answer is that it is continuous but I just wanted somebody to confirm why.

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Nov 15, 2011

mathman

Science Advisor

Since yo is fixed, you simply have a function of one variable (x), so apply what you know about a function of one variable and its derivative.

1. What is continuity in two-variable functions?

Continuity in two-variable functions refers to the property of a function where small changes in the input values result in small changes in the output values. In other words, if we can draw a smooth curve without any breaks or jumps to represent the function, then it is considered continuous.

2. How is continuity different from differentiability?

Continuity and differentiability are closely related but distinct concepts. While continuity only requires small changes in input values to result in small changes in output values, differentiability also requires the existence of a derivative at every point in the domain of the function.

3. Can a function be continuous but not differentiable?

Yes, a function can be continuous but not differentiable. This can happen when there is a sharp turn or corner in the graph of the function, which results in a break in the slope and thus, the function is not differentiable at that point.

4. How can we determine continuity of a two-variable function?

To determine continuity of a two-variable function, we can use the following three conditions: 1) The function must be defined at the point in question, 2) The limit of the function as it approaches the point from both directions must exist and be equal, 3) The value of the function at the point must be equal to the limit.

5. Why is continuity important in mathematics and science?

Continuity is important in mathematics and science because it allows us to make predictions and draw conclusions about a function without having to evaluate every single point. It also helps us to understand the behavior of a function and its graph, which is essential in many areas of science and engineering.