Problem 30 - 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 differentiable, then it is not continu- ous.

Step-by-Step Solution

Answer

The statement is false; a counterexample is \( f(x) = |x| \) which is continuous but not differentiable at \( x = 0 \).

Step 1: Understand the Problem

We need to determine whether the statement "If a function is not differentiable, then it is not continuous" is true or false. This involves evaluating the relationship between differentiability and continuity.

Step 2: Recall Definitions

A function is differentiable at a point if it has a derivative there, meaning the slope of the tangent line is defined. A function is continuous at a point if there is no break, jump, or hole at that point.

Step 3: Assess the Statement

According to mathematical principles, differentiability implies continuity, but the converse is not true. Therefore, a function can be continuous but not differentiable at a certain point.

Step 4: Provide a Counterexample

Consider the function \( f(x) = |x| \). It is continuous everywhere, specifically at \( x = 0 \), but it is not differentiable at that point because the slope from the left does not equal the slope from the right (it has a corner).

Step 5: Decide the Truth Value

Since we found a function (\( f(x) = |x| \)) that is continuous but not differentiable at a point, the statement "If a function is not differentiable, then it is not continuous" is false.

Key Concepts

Function Continuity

Understanding function continuity is crucial for comprehending the behavior of functions in calculus. A function is said to be continuous at a point if there is no interruption in its graph at that point. Imagine drawing the graph of the function without lifting your pen; if you can, the function is continuous. In mathematical terms, a function \( f(x) \) is continuous at a point \( x = a \) if the limit of \( f(x) \) as \( x \) approaches \( a \) is equal to the function's value at \( a \). This is expressed as:

• \( \lim_{x \to a} f(x) = f(a) \)

A function that is continuous over an entire interval does not have any breaks, jumps, or discontinuities within that interval. Continuity is a necessary condition for differentiability but not a guarantee. Not all continuous functions have a well-defined derivative everywhere, which makes this concept a stepping stone to understanding differentiability.

Differentiable Functions

Differentiability is a deeper concept than continuity in calculus, focusing on how a function behaves at a more granular level. A function is differentiable at a point if it has a derivative at that point. This means the function has a well-defined tangent line at that point, and its graph is smooth there. The derivative represents the rate of change or slope of the function at that given point.

• If a function \( f(x) \) is differentiable at a point \( x = a \), it must first be continuous at \( x = a \).
• However, the opposite is not always true: a function can be continuous but not differentiable at a point.

Taking the function \( f(x) = |x| \) as an example helps to understand this. The function is continuous at \( x = 0 \), but not differentiable because there is a sharp corner at that point. Calculus tells us that where the graph has corners or cusps, differentiability fails, although continuity may still exist.

Counterexample in Calculus

Counterexamples are powerful tools in calculus for disproving general statements. When faced with a proposition such as "If a function is not differentiable, then it is not continuous," providing a counterexample can show the claim is false. One classic counterexample is the function \( f(x) = |x| \), which is continuous everywhere but not differentiable at \( x = 0 \).

• This example shows how differentiability is stricter than continuity; continuity does not require the smoothness and well-defined tangents demanded by differentiability.
• In visual terms, a counterexample like \( f(x) = |x| \) demonstrates a point on the graph where the pants meet without a clear slope direction, undermining any concluding universal differentiability.
• Such examples are crucial in math for demonstrating exceptions and understanding the boundaries of mathematical rules fully.

Using counterexamples helps students and mathematicians alike to deepen their insights into how and why mathematical properties hold and the nuances between closely related concepts like continuity and differentiability.