Problem 94 - Single Variable Calculus

Question

Prove that if \(f\) and \(g\) are continuous at \(a\), then \(f-g\) is continuous at \(a\).

Step-by-Step Solution

Answer

To prove that if f and g are continuous at a, then (f-g) is also continuous at a, we define: \((f-g)(x) = f(x) - g(x)\). Using the properties of limits and knowing that f and g are continuous at a, we can find the limit as x approaches a: \[\lim_{x \to a} (f-g)(x) = \lim_{x \to a} (f(x) - g(x)) = \lim_{x \to a} f(x) - \lim_{x \to a} g(x) = f(a) - g(a)\] Since \(\lim_{x \to a} (f-g)(x) = (f-g)(a)\), we can conclude that (f-g)(x) is continuous at point a.

Step 1: Define Continuity

A function h(x) is continuous at point a if the following condition is met: \[\lim_{x \to a} h(x) = h(a)\] So we are given that f and g are continuous at a, meaning: \[\lim_{x \to a} f(x) = f(a)\] and \[\lim_{x \to a} g(x) = g(a)\]

Step 2: Define the Function (f-g)(x)

We will define the function (f-g)(x), which is essentially subtracting g(x) from f(x): \[(f-g)(x) = f(x) - g(x)\]

Step 3: Find the Limit of (f-g)(x) as x Approaches a

To show that the function (f-g)(x) is continuous at a, we need to find the limit of this function as x approaches a: \[\lim_{x \to a} (f-g)(x) = \lim_{x \to a} (f(x) - g(x))\]

Step 4: Use the Properties of Limits

We know that the limit of a sum or difference of functions is the sum or difference of their limits, assuming both limits exist. Therefore, we can write: \[\lim_{x \to a} (f(x) - g(x)) = \lim_{x \to a} f(x) - \lim_{x \to a} g(x)\]

Step 5: Substitute in the Continuity Condition

Since we know that f and g are continuous at a, we can write: \[\lim_{x \to a} f(x) - \lim_{x \to a} g(x) = f(a) - g(a)\]

Step 6: Connect the Evaluation of the Limit to Continuity

We now have: \[\lim_{x \to a} (f-g)(x) = f(a) - g(a)\] But we know that: \[(f-g)(a) = f(a) - g(a)\] Thus, by the definition of continuity, we have: \[\lim_{x \to a} (f-g)(x) = (f-g)(a)\]

Step 7: Conclusion

Since we have shown that the limit of (f-g)(x) as x approaches a is equal to (f-g)(a), the difference between f and g, (f-g)(x), is continuous at point a.

Key Concepts

Limit of a Function

Understanding the concept of a limit is essential when dealing with continuity in calculus. A limit describes the value that a function approaches as its input (or 'x' value) gets closer to a certain point. Mathematically, the expression
\(lim_{x \to a} f(x) = L\)
implies that as x gets arbitrarily close to 'a', the function f(x) approaches the value L. The limit does not concern itself with the value of the function at 'a', but rather with the behavior of the function as it gets close to 'a'.

The concept of a limit is the cornerstone in defining continuity at a point, as a function is said to be continuous at a point 'a' if and only if
\(lim_{x \to a} f(x) = f(a)\).
In essence, if the limit exists at that point and equals the function's value at that point, the function can be thought of as having no gaps, jumps, or breaks there.

Properties of Limits

Limits follow specific rules that aid in their calculation. These properties ensure that the process of finding limits is consistent and can be applied to a wide variety of functions. Some of the fundamental properties of limits in calculus include

Sum and Difference Rule

The limit of a sum or difference is equal to the sum or difference of the limits, assuming the limits exist individually. In other words,
\(lim_{x \to a} (f(x) \pm g(x)) = \lim_{x \to a} f(x) \pm \lim_{x \to a} g(x)\).

Product Rule

The limit of a product is the product of the limits,
\(lim_{x \to a} [f(x)g(x)] = \lim_{x \to a} f(x) \cdot \lim_{x \to a} g(x)\).

Quotient Rule

The limit of a quotient is the quotient of the limits, provided the limit of the denominator is not zero,
\(lim_{x \to a} \frac{f(x)}{g(x)} = \frac{\lim_{x \to a} f(x)}{\lim_{x \to a} g(x)}\), with \(\lim_{x \to a} g(x) eq 0\).

These properties play a crucial role in finding the limit of more complex functions and in proving the continuity of functions as exemplified in the given exercise.

Difference of Continuous Functions

The difference of two continuous functions is a concept that can be made simpler when we understand that continuity is preserved through basic operations. When given two functions, \(f\) and \(g\), that are both continuous at a certain point 'a', their difference
\((f-g)(x) = f(x) - g(x)\)
is also continuous at that same point. This happens because the properties of limits ensure that the limit of a difference is the difference of the limits. Hence, if each individual function does not have any interruptions or breaks at 'a', their difference will inherit this smooth behavior.

In the context of the textbook exercise, the students must show that for the function difference \((f-g)(x)\), the limit as x approaches a, is the difference of the function values at 'a'. Therefore, if both \(f(x)\) and \(g(x)\) are continuous at 'a', the resulting function \((f-g)(x)\) will be continuous there as well.

Continuity Condition

The continuity condition is the formal criterion that defines whether a function is continuous at a particular point. For a function to be continuous at a point 'a', three conditions must be met:

• The function must be defined at 'a'.
• The limit of the function as x approaches 'a' must exist.
• The limit of the function as x approaches 'a' must equal the function's value at 'a'.

This is succinctly captured by the equation
\(lim_{x \to a} f(x) = f(a)\).
These criteria ensure that there is no 'break' in the graph of the function at the point, allowing the function to be smooth and unbroken as it passes through that point. The continuity condition also extends beyond a single point to an interval, where a function is continuous on an interval if it is continuous at every point in that interval.

The continuity condition not only provides the basis for mathematical proofs, as in the textbook exercise, but also holds significant implications in real-world situations where continuous behavior of a function aligns with observed natural phenomena.