Question

We informally describe a function \(f\) to be continuous at \(a\) if its graph contains no holes or breaks at \(a\). Explain why this is not an adequate definition of continuity.

Short answer

Answer: The informal definition of continuity is not adequate because it relies only on the visual aspect of the function and does not consider the precise mathematical behavior of the function at and around a point. The formal definition of continuity provides a more thorough analysis of a function's behavior and gives a better indication of whether the function is truly continuous or not.

Step-by-step solution

Understand the informal and formal definitions of continuity

First, let's look at both the informal and formal definitions of continuity. The informal definition given in the exercise states that a function is continuous at a point \(a\) if its graph contains no holes or breaks at \(a\). On the other hand, the formal definition for a function, \(f(x)\), to be continuous at a point \(a\) is as follows: A function \(f(x)\) is continuous at \(a\) if: 1. \(f(a)\) is defined (i.e., the function has a value at point \(a\)), 2. \(\lim_{x \to a} f(x)\) exists, and 3. \(\lim_{x \to a} f(x) = f(a)\).

Compare the informal and formal definitions

Comparing the informal definition to the formal one, we can see that the informal definition focuses solely on the visual aspect of the function, specifically looking for holes or breaks in the graph at point \(a\). While this can sometimes give a general idea of whether a function is continuous or not, it doesn't take into account more precise mathematical behaviors. The formal definition, on the other hand, focuses on the function's behavior as \(x\) approaches \(a\) and whether the limit as \(x\) approaches \(a\) is equal to the function's value at point \(a\). This allows us to consider not only what happens at the specific point but also how the function behaves around it.

Providing examples illustrating the inadequacy of the informal definition

Now, let's consider an example where the informal definition would fall short: Consider the function \(f(x) = \frac{x^2 - 1}{x - 1}\). This function has a hole at \(x = 1\). However, if we simplify the function, we get: \(f(x) = \frac{(x - 1)(x + 1)}{x - 1} = x + 1\) when \(x \neq 1\) This would imply that the function is continuous for all \(x\) except for \(x = 1\), but if we plug in \(x = 1\) after cancelling the \((x - 1)\) terms, we see that \(f(1) = 1 + 1 = 2\). In the formal definition, this would show that the function is continuous at \(x = 1\) (since the limit exists and is equal to the function's value at \(x = 1\)), but, from the informal point of view, we would still see a hole in the graph and consider it discontinuous. This example illustrates that the informal definition of continuity is not adequate, as it does not take into account all the mathematical behaviors at and around a point - something that the formal definition does. In conclusion, the informal definition of continuity given in the exercise is not adequate because it relies only on the visual aspect of the function and does not consider the precise mathematical behavior of the function at and around a point. The formal definition of continuity provides a more thorough analysis of a function's behavior and gives a better indication of whether the function is truly continuous or not.