Problem 40 - Calculus with Concepts in Calculus

Question

Show that if \(f\) is differentiable at \(\left(x_{0}, y_{0}\right)\), then \(f\) is continuous at \(\left(x_{0}, y_{0}\right) .\) (Hint: Using (7), show that \(\left.\lim _{(x, y) \rightarrow\left(x_{0}, y_{0}\right)} f(x, y)=f\left(x_{0}, y_{0}\right) .\right)\)

Step-by-Step Solution

Answer

If \( f \) is differentiable at \( (x_0, y_0) \), it is also continuous at that point.

Step 1: Understand Differentiability

To show that a function is continuous at a point, we first need to understand the implications of differentiability. A function of two variables, \( f(x, y) \), is differentiable at \( (x_0, y_0) \) if the following equation holds: \[ f(x_0 + h, y_0 + k) = f(x_0, y_0) + A \, h + B \, k + \ \text{error terms} \ \text{where error terms} = o(\sqrt{h^2 + k^2}) \] as \( (h, k) \to (0, 0) \). This means the error terms go to 0 faster than \( \sqrt{h^2 + k^2} \).

Step 2: Write the Continuity Condition

For continuity at \( (x_0, y_0) \), we need to show that \[ \lim_{(x, y) \to (x_0, y_0)} f(x, y) = f(x_0, y_0). \] This requires that the function values at \( (x, y) \) approach the function value at \( (x_0, y_0) \) as \( (x, y) \) approaches \( (x_0, y_0) \).

Step 3: Express the Limit Using Differentiability

Substitute \( x = x_0 + h \) and \( y = y_0 + k \) into the differentiability equation, which gives: \[ f(x, y) = f(x_0, y_0) + A \, h + B \, k + o(\sqrt{h^2 + k^2}). \] To find the limit, consider \( (h, k) \to (0, 0) \), then \[ \lim_{(h, k) \to (0, 0)} \left[ A \, h + B \, k + o(\sqrt{h^2 + k^2}) \right] = 0, \] because the linear terms will disappear into 0 and the error term \( o(\sqrt{h^2 + k^2}) \) also vanishes as \( (h, k) \to (0, 0) \).

Step 4: Confirm Continuity Conclusion

Since \( \lim_{(x, y) \to (x_0, y_0)} \left[f(x_0, y_0) + A \, h + B \, k + o(\sqrt{h^2 + k^2})\right] = f(x_0, y_0) \), it follows that \[ \lim_{(x, y) \to (x_0, y_0)} f(x, y) = f(x_0, y_0). \] Therefore, \( f \) is continuous at \( (x_0, y_0) \).

Key Concepts

Continuity

Continuity is a fundamental concept in calculus. It essentially ensures that there are no sharp jumps or breaks in the graph of a function at a specific point. For a function of two variables, like our function \( f(x, y) \), continuity at a point \( (x_0, y_0) \) means that as we take points \( (x, y) \) arbitrarily close to \( (x_0, y_0) \), the function values \( f(x, y) \) also get arbitrarily close to \( f(x_0, y_0) \).

This can be expressed mathematically as the limit:

• \( \lim_{(x, y) \to (x_0, y_0)} f(x, y) = f(x_0, y_0) \).

Continuity is essential for differentiability, as a function must be continuous at a point to be differentiable there. However, it is important to note that continuity alone does not guarantee differentiability. Differentiability is a stronger condition that must first satisfy continuity.

Multivariable Calculus

In multivariable calculus, we extend concepts from single-variable calculus to functions of more than one variable. Here, we deal with functions like \( f(x, y) \), which depend on multiple inputs.

Differentiability is one of these concepts that extends to several variables. For a function \( f(x, y) \) to be differentiable at \( (x_0, y_0) \), the function should closely approximate a plane at that point. This is represented by the equation:

• \( f(x_0 + h, y_0 + k) = f(x_0, y_0) + A \, h + B \, k + \text{error terms} \).

The error terms become negligible faster than the distance \( \sqrt{h^2 + k^2} \) approaches zero.

Multivariable calculus allows us to analyze complex systems and models that involve several variables. Mastery of these concepts is crucial for advanced studies in physics, engineering, and other applied sciences.

Limit Definition

Understanding limits is crucial when diving into calculus and especially when proving continuity and differentiability. The concept of a limit helps us describe the value that a function approaches as its inputs get closer to a given point.

To show continuity or differentiability, limits are used to express how the function behaves near a certain point. For example, the statement \( \lim_{(x, y) \to (x_0, y_0)} f(x, y) = f(x_0, y_0) \) means that as \( x \) and \( y \) approach \( x_0 \) and \( y_0 \) respectively, \( f(x, y) \) approaches \( f(x_0, y_0) \).

• The limit is a tool for identifying small-scale behavior.
• It reveals how a function smoothly bridges behavior around points.
• Using a limit ensures that no abrupt changes happen at the convergence point.

Applying the concept of limits within differentiability confirms that error terms vanish, seamlessly proving continuity in the context of multivariable calculus. It's a delicate balance that showcases the intricate structure of calculus.