Problem 55 - Calculus for Biology and Medicine

Question

If \(f(x)\) is differentiable for all \(x \in \mathbf{R}\) except at \(x=c\), is it true that \(f(x)\) must be continuous at \(x=c\) ? Justify your answer.

Step-by-Step Solution

Answer

No, \(f(x)\) is not necessarily continuous at \(x = c\) if it is not differentiable there.

Step 1: Understand Differentiability and Continuity

A function is said to be differentiable at a point if its derivative exists at that point. A function is continuous at a point if the limit of the function as it approaches the point equals the function's value at the point, i.e., \( \lim_{{x \to c}} f(x) = f(c) \). Differentiability implies continuity, but continuity does not imply differentiability.

Step 2: Consider the Given Condition

The function \( f(x) \) is differentiable for all \( x \in \mathbb{R} \) except at \(x = c\). This means \( f(x) \) is not differentiable at \(x = c\), which implies it may not be continuous at \(x = c\) either since differentiability implies continuity.

Step 3: Provide an Example

Consider \(f(x) = |x|\). \(f(x)\) is differentiable everywhere except at \(x = 0\). At \(x = 0\), the left-hand derivative and the right-hand derivative do not match, so it is not differentiable at this point. However, \(f(x)\) is continuous at \(x = 0\) because \( \lim_{{x \to 0}} |x| = 0 = |0|\).

Step 4: Conclusion

The example shows that a function \(f(x)\) can be continuous at a point where it is not differentiable. Therefore, although differentiability implies continuity, if a function is not differentiable at a point, it might still be continuous at that point.

Key Concepts

Differentiability

Differentiability is the property of a function that signifies whether its derivative exists at a particular point. In simpler terms, a function is differentiable at a point if you can draw a tangent line to its graph at that point.

• This implies that the function has a specific, well-defined slope at that point.
• If a function is differentiable at every point in its domain, it is said to be globally differentiable within that domain.

A common misconception is that differentiability automatically leads to continuity, which is true. However, the opposite—continuity leading to differentiability—is not true. For example, some functions may have sharp turns or cusps, like the absolute value function at its vertex, where the function is continuous but not differentiable due to the absence of a unique tangent. Understanding this concept provides insight into why the behavior of functions can vary significantly even in regions where they appear smooth.

Limit

The concept of a limit is foundational in understanding continuity and differentiability. Limit involves approaching a particular point from all possible directions and seeing where the function's values tend to go.

• The limit exists if, as you get infinitely close to a point, the values of the function approach a specific number.
• For a function to be continuous at a point, the function's limit as it approaches that point must equal the actual function value at the point.

Mathematically, a limit can be represented as: \[ \lim_{{x \to c}} f(x) = L \]If the left-hand and right-hand limits both exist and are equal as they approach the same point from opposite directions, then the overall limit exists. Limits can be thought of as the stepping stones towards grasping ideas such as continuity and differentiability. In the exercise, we saw how a discontinuity in differentiability at a certain point does not necessarily mean a discontinuity in terms of the function's value.

Derivative

The derivative of a function at a point provides the rate at which the function is changing at that specific point. This is often visualized as the slope of the tangent line to the function's graph at that point.

• The derivative, denoted as \( f'(x) \), represents the instantaneous rate of change of the function with respect to the variable \( x \).
• Having a derivative means the graph of the function has no sharp turns, vertical tangents, or discontinuities at that point.

The connection between continuity and differentiability is tied to the derivative. For example, if a function \( f(x) \) is differentiable at \( x = c \), then it must be continuous at \( x = c \). However, the converse is not always true, as demonstrated with the absolute value function where the slope abruptly changes direction, leading to a dearth of derivative despite the absence of a gap in the function itself. Understanding derivatives helps us not only in analyzing rates of change but also in comprehending the deeper properties of function continuity.