Problem 55 - Calculus Early Transcendentals

Question

Prove that \(f\) is continuous at a if and only if $$\lim _{h \rightarrow 0} f(a+h)=f(a)$$

Step-by-Step Solution

Answer

The limit condition is equivalent to the definition of continuity at \( a \).

Step 1: Understanding continuity at a point

A function \( f \) is said to be continuous at a point \( a \) if the following condition holds: \( \lim_{x \to a} f(x) = f(a) \). This means the limit of \( f(x) \) as \( x \) approaches \( a \) should equal the value of the function at \( a \).

Step 2: Rewriting the limit definition

We need to show the equivalence of \( \lim_{h \to 0} f(a+h) = f(a) \) with continuity at \( a \). Start by noticing that \( h \to 0 \) means \( a+h \to a \). Thus, \( \lim_{h \to 0} f(a+h) = \lim_{x \to a} f(x) \).

Step 3: Proving the 'if' part

Assume \( \lim_{h \to 0} f(a+h) = f(a) \). By definition, as \( h \to 0 \), \( a+h \) approaches \( a \), hence \( \lim_{x \to a} f(x) = f(a) \). This implies that \( f \) is continuous at \( a \) by the definition of continuity.

Step 4: Proving the 'only if' part

Assume \( f \) is continuous at \( a \) which means \( \lim_{x \to a} f(x) = f(a) \). Hence, with \( x = a + h \), as \( h \to 0 \), \( x \to a \), implying \( \lim_{h \to 0} f(a+h) = \lim_{x\to a} f(x) = f(a) \).

Step 5: Concluding proof

Both the 'if' and 'only if' parts are proved showing that the two conditions are equivalent. Thus, the limit \( \lim_{h \to 0} f(a+h) = f(a) \) holds if and only if \( f \) is continuous at \( a \), completing the proof.

Key Concepts

Limit of a function

Understanding the limit of a function is crucial in calculus because it describes the behavior of a function as its input approaches a certain point. Let's say we have a function \( f(x) \). The limit \( \lim_{x \to a} f(x) \) essentially asks, "What value does \( f(x) \) get closer to as \( x \) gets closer to \( a \)?" This concept helps us handle functions at points where they might not be well-defined.

When calculating limits, some functions can approach a particular value smoothly, while other times, they might oscillate or diverge. Thus, even without knowing \( f(a) \) directly, if we understand how \( f(x) \) behaves as it nears \( a \), we can determine the limit. This becomes fundamental in analyzing the continuity and differentiability of functions.

A common challenge is when a function is defined differently at a point than it is everywhere else. Limits allow us to formally understand and calculate these scenarios. This foundational idea sets the stage for exploring continuity and other dynamic behaviors of functions as they transition smoothly from one state to another.

Definition of continuity

Continuity is a key idea in calculus, which intuitively means a function has no breaks, jumps, or holes. Formally, a function \( f \) is said to be continuous at a point \( a \) if \( \lim_{x \to a} f(x) = f(a) \). This essentially means the behavior of \( f \) as \( x \) nears \( a \) is predictable and matches its actual value at \( a \).

There are three main conditions for continuity at a point:

• \( f(a) \) is defined, meaning that \( f \) has a value at \( a \).
• The limit \( \lim_{x \to a} f(x) \) exists.
• \( \lim_{x \to a} f(x) = f(a) \).

If all these conditions are satisfied, the function does not have any abrupt changes around that point.

Understanding this definition helps students see how functions behave in real-life applications where a continuous transition from one state to another is essential. This understanding is particularly useful when working with graphs, as it assures that there are no sudden jumps or undefined points within the plotted curve.

Pointwise continuity

The notion of pointwise continuity emphasizes checking the continuity of a function at each point in its domain individually. For a function to be globally continuous over its domain, it is sufficient to be continuous at all points where it is defined.

To assess pointwise continuity, you examine the behavior of a function at every point \( a \) in its domain. For each point, you verify that the conditions for continuity (the three conditions mentioned earlier) hold true. If this is the case for all points, you can conclude that the function is continuous everywhere within its domain.

In practice, when dealing with more complex functions, this approach becomes useful to ensure that no point has been overlooked. Sometimes, functions are pieced together using different formulas for different intervals, leading to the necessity for pointwise evaluation.

• The concept of pointwise continuity allows mathematicians to analyze functions that are comprised of multiple parts.
• This involves scrutinizing each segment of the function to ensure smooth transitions at the boundaries.

This detailed analysis ensures understanding and efficiently detecting points of discontinuity, if any, allowing for correcting or adapting the function's behavior as needed.