Problem 51 - Calculus for Biology and Medicine

Question

Which of the following statements is true? (A) If \(f(x)\) is continuous, then \(f(x)\) is differentiable. (B) If \(f(x)\) is differentiable, then \(f(x)\) is continuous.

Step-by-Step Solution

Answer

Statement (B) is true: Differentiability implies continuity.

Step 1: Understanding Differentiability and Continuity

Recall that if a function is differentiable at a point, it is also continuous at that point. In mathematical terms, if \( f(x) \) is differentiable at \( x = a \), then it is continuous at \( x = a \). This means if \( f'(a) \) exists, then \( \lim_{{x \to a}} f(x) = f(a) \).

Step 2: Analyzing Statement (A)

Statement (A) claims that if \( f(x) \) is continuous, then \( f(x) \) is differentiable. However, continuity does not imply differentiability. A common counterexample is \( f(x) = |x| \). This function is continuous everywhere but not differentiable at \( x = 0 \).

Step 3: Analyzing Statement (B)

Statement (B) claims that if \( f(x) \) is differentiable, then \( f(x) \) is continuous. As reviewed in Step 1, differentiability does imply continuity. Therefore, if \( f(x) \) is differentiable at a point, it is also continuous at that point.

Key Concepts

Differentiability

Differentiability is a key concept in calculus that examines how smoothly a function behaves. A function is said to be differentiable at a point if its derivative exists at that point. This means the tangent to the curve at that point is well-defined and finite. When we talk about the derivative, we're talking about the instantaneous rate of change of the function at that point. To visualize this, imagine the slope of the tangent line at a specific point on the curve; that's essentially what the derivative tells us.
Differentiability means:

• The function must possess a derivative at the given point, which implies the slope of the tangent does not have any jumps or breaks.
• If a function is differentiable at a point, it must also be continuous there. Therefore, differentiability implies continuity, but not the other way around.

A classic example is the absolute value function, \(f(x) = |x|\), which is not differentiable at \(x = 0\) even though it is continuous there.

Continuity

Continuity is at the heart of understanding how a function behaves without interruptions. A function is said to be continuous at a point if it satisfies three main conditions:

• The function is defined at that point.
• The limit of the function as it approaches that point exists.
• The value of the function at that point equals the limit value.

In simpler terms, you should be able to draw the graph of a continuous function without lifting your pencil from the paper.
Continuity can be understood in terms of the limit behavior of the function near a point. If you find that \(\lim_{{x \to a}} f(x) = f(a)\), then the function satisfies the condition for continuity at \(x = a\). However, not every continuous function is differentiable. Again, the absolute value function \(f(x) = |x|\) ensures continuity but fails the differentiability test at zero.

Limits

Limits are foundational in calculus, acting as the bridge between intuitive notions of function behavior and precise calculations. At its core, a limit describes what the value of a function approaches as the input gets closer to a certain point. It's written mathematically as \(\lim_{{x \to a}} f(x)\).
Limits:

• Allow us to formally define both continuity and differentiability.
• Help evaluate the behavior of functions, even for values that may not be directly computable initially.

Consider the limit of a function as \(x\) approaches \(a\): if both the left-hand limit and the right-hand limit as \(x\) approaches \(a\) are equal, we can say the two-sided limit exists.
This key idea makes it possible to discuss the continuity of functions effectively. Without limits, the precise mathematical understanding of both continuous and differentiable functions wouldn’t be possible.