Problem 29 - Calculus

Question

Are the statements true or false? If a statement is true, give an example illustrating it. If a statement is false, give a counterexample. If a function is not continuous, then it is not differentiable.

Step-by-Step Solution

Answer

True; non-continuity implies non-differentiability, as shown by a jump.

Step 1: Understanding the Concepts

To determine the truth of the statement, we need to understand two concepts: continuity and differentiability. A function is continuous at a point if it does not have any breaks, jumps, or holes at that point. A function is differentiable at a point if it has a defined tangent, implying it is smooth enough without any sharp turns at that point.

Step 2: Analyzing the Statement

The statement says that if a function is not continuous, then it is not differentiable. Differentiability implies continuity; therefore, if a function is not continuous at a point, it cannot be differentiable at that point. This supports the statement's validity.

Step 3: Giving an Example

Consider the function \[f(x) = \begin{cases} x^2, & x eq 2 \ 5, & x = 2\end{cases}\]This function is not continuous at \(x = 2\) because there is a jump from \(4\) to \(5\). Since the function is not continuous at \(x = 2\), it is also not differentiable at that point.

Key Concepts

Continuity

In calculus, continuity is a fundamental concept that helps determine if a function behaves smoothly at a particular point. A function is considered continuous at a point if there are no interruptions in its graph at that point. This means no gaps, jumps, or abrupt changes should be present.
To verify if a function is continuous at a certain point, you should check:

• The function is defined at the point.
• The limit of the function as it approaches the point is the same from both the left and right sides.
• Both the limit of the function and its actual value at the point are equal.

If all these conditions are satisfied, the function is continuous at that point. For example, the function \(f(x) = x^2\) is continuous at every point because it meets these criteria for any \(x\).
Understanding continuity is crucial because it underlies more complex concepts such as differentiability, which we will explore next.

Differentiability

Differentiability is a property of a function that indicates whether it has a well-defined tangent line at a given point. If a function is differentiable at a point, it means the function has a smooth slope without any breaks or sharp turns.
For a function to be differentiable,

• It must first be continuous at the point in question.
• The function must have a defined derivative—that is, the slope of the tangent must exist at that point.
• No vertical tangent lines or cusps should occur at the point.

A differentiable function is always continuous at the point, but as shown in some examples, a continuous function is not always differentiable. Consider the absolute value function \(f(x) = |x|\), which is continuous everywhere but not differentiable at \(x = 0\) because it has a sharp corner there.
Thus, differentiability requires an additional condition of smooth curvature beyond just continuity.

Counterexample

A counterexample is a specific instance or example that proves a general statement or proposition false. In math, counterexamples are crucial in testing the validity of statements. They show that assumptions which might seem logical are not always true.
When we say, "If a function is not continuous, then it is not differentiable," a counterexample is not needed because this statement is inherently true. Differentiability requires continuity, so if a function breaks continuity, it inherently lacks differentiability.
On the other hand, if we reverse the statement to: "If a function is continuous, it is differentiable," we encounter instances where this is false. The absolute value function provides a counterexample for this reversed statement.
Counterexamples like these help clarify understandings in calculus by offering concrete proof against assumed truths, guiding students to approach problems with critical thinking.