Question

True or False? Determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. If \(f(x)=g(x)\) for \(x \neq c\) and \(f(c) \neq g(c),\) then either \(f\) or \(g\) is not continuous at \(c\).

Short answer

The statement is true.

Step-by-step solution

Understanding Continuity

By definition, a function \(f\) is continuous at a point \(x=c\) if and only if the following three conditions are satisfied: 1. \(f(c)\) is defined. 2. \(\lim_{{x \to c}} f(x)\) exists. 3. \(\lim_{{x \to c}} f(x) = f(c)\). Therefore, a function is discontinuous at a point if any of these conditions is violated.

Applying the Definition to The Given Case

In the given problem, \(f(x)=g(x)\) for \(x \neq c\) and \(f(c) \neq g(c)\). It implies that \(\lim_{{x \to c}} f(x) = \lim_{{x \to c}} g(x)\) as per the given condition, \(f(x) = g(x)\) for all \(x \neq c\). But the limits of both functions at \(x=c\) are not equal to their function values at \(x=c\), since \(f(c) \neq g(c)\). Hence, we can conclude that either function \(f\) or \(g\) is discontinuous at \(c\), or potentially both of them are.

Conclusion

So the statement 'If \(f(x)=g(x)\) for \(x \neq c\) and \(f(c) \neq g(c)\), then either \(f\) or \(g\) is not continuous at \(c\)' is true. The situation described ensures that either \(f\) or \(g\) – or possibly both, doesn't satisfy all the conditions for a function to be continuous at a point, i.e., they are discontinuous at \(c\).

Key concepts

Limits of Functions

The concept of a limit is foundational in calculus and pertains to the behavior of functions as they approach a particular point. Limits help us understand the value that a function approaches as the input approaches a certain value. In mathematical terms, the limit of a function f(x) as x approaches a value c is expressed as \( \lim_{{x \to c}} f(x) \).

For instance, consider a function that gets closer to the value 5 as x approaches 3. In this case, we would say \( \lim_{{x \to 3}} f(x) = 5 \). To truly understand if the function reaches this limit, we often look from both sides: as x approaches 3 from the left (denoted as \( x \to c^- \)) and from the right (denoted as \( x \to c^+ \)). If these two one-sided limits are equal, the limit of the function exists at that point.

It's crucial to note that a function does not need to be defined at x = c for a limit to exist there; what matters is the function's approach as x gets arbitrarily close to c. This subtlety is crucial for the understanding of both continuity and discontinuity in functions.

Continuity of Functions

Continuity of a function at a point is rooted in the seamless, unbroken nature of the function's graph at that point. For a function f(x) to be continuous at x = c, three conditions must be met, as explained in the exercise:

1. The function must be defined at x = c, meaning f(c) exists.
2. The limit of the function as x approaches c must exist.
3. The function's value at c must equal the limit as x approaches c, symbolically, \( \lim_{{x \to c}} f(x) = f(c) \).

Understanding Continuity Through Examples

A polynomial function like f(x) = x^2 is continuous for all real numbers because at every point c, the above three conditions hold true. Importantly, if a function meets the continuity criteria at every point within a certain interval, we refer to the function as being continuous on that interval.

Remember, if any of these conditions are not satisfied, we no longer have continuity. The exercise demonstrates that even if the values of two functions match up everywhere except for one point, the mismatch at that single point can disrupt continuity, which is a concept that often confounds students.

Discontinuous Functions

Discontinuous functions have disruptions in their graphs at one or more points. Here, the smooth behavior of the function is interrupted, leading to a break, jump, or an undefined point. There are various types of discontinuities, each characterized by the failure to meet the conditions for continuity:

• Point Discontinuity: This occurs when a function f(x) is not continuous at some point x = c. However, the limit may exist when approaching from both sides, like in a piecewise function with a defined jump at c.
• Jump Discontinuity: Here, the one-sided limits exist but are not equal to each other as x approaches c, leading to a sudden 'jump' in the function's graph.
• Infinite Discontinuity: This type happens when the limit approaches infinity as x approaches c, often visualized as a vertical asymptote in the graph of f(x).

In the textbook exercise provided, we're presented with a case of point discontinuity at x = c where the two functions f(x) and g(x) differ only at one point. Even though the functions agree on every other value, this single disagreement leads to a discontinuity, reinforcing the idea that continuity requires adherence to the strict conditions at every single point within the domain of the function.