Continuous But Not Differentiable Piecewise Functions At X=-3

In the realm of calculus, the concepts of continuity and differentiability are fundamental. A function is said to be continuous at a point if there is no break in the graph at that point, meaning the limit exists, the function is defined, and the limit equals the function's value. Differentiability, on the other hand, is a stricter condition. A function is differentiable at a point if its derivative exists at that point, implying the function must be continuous and possess a well-defined tangent line. This article delves into piecewise functions, exploring how to determine whether they are continuous and differentiable at a specific point. Specifically, we will address the question of identifying a piecewise function that is continuous at x = -3 but not differentiable at the same point. This scenario highlights an interesting characteristic of functions, where continuity does not necessarily guarantee differentiability. Through a detailed analysis, we will unravel the conditions that must be met for a piecewise function to exhibit this behavior, providing a comprehensive understanding of these critical calculus concepts.

Before we dive into the specifics of piecewise functions, it's crucial to have a solid understanding of continuity and differentiability.

• Continuity: A function f(x) is continuous at a point x = a if the following three conditions are met:

  1. f(a) is defined.
  2. The limit of f(x) as x approaches a exists.
  3. The limit of f(x) as x approaches a is equal to f(a).

  In simpler terms, a function is continuous if you can draw its graph without lifting your pen from the paper. There are no sudden jumps or breaks.

• Differentiability: A function f(x) is differentiable at a point x = a if the derivative f'(a) exists. The derivative represents the instantaneous rate of change of the function at that point, which geometrically corresponds to the slope of the tangent line. For a function to be differentiable at a point, it must first be continuous at that point. However, continuity is not sufficient for differentiability. A function can be continuous but not differentiable if it has a sharp corner, a cusp, or a vertical tangent at that point. The function must be smooth at the point in question for the derivative to exist. The left-hand and right-hand derivatives must exist and be equal at the point for the function to be differentiable.

The relationship between continuity and differentiability can be summarized as follows: differentiability implies continuity, but continuity does not imply differentiability. This means that if a function is differentiable at a point, it is also continuous at that point. However, if a function is continuous at a point, it may or may not be differentiable at that point. Understanding this nuance is essential for analyzing the behavior of functions, especially piecewise functions.

A piecewise function is a function defined by multiple sub-functions, where each sub-function applies to a specific interval of the domain. These functions are crucial in modeling real-world scenarios where different rules apply under different conditions. Piecewise functions can exhibit interesting behaviors at the points where the sub-functions meet, often called "breakpoints." At these breakpoints, the function's continuity and differentiability must be carefully examined. A piecewise function is a function that is defined by multiple sub-functions, each applying to a specific interval of the domain. For example, consider the following piecewise function:

f(x) = 
  \begin{cases}
    x^2, & \text{if } x < 0 \\
    x, & \text{if } 0 \le x \le 1 \\
    1, & \text{if } x > 1
  \end{cases}

This function is defined by three sub-functions: x² for x < 0, x for 0 ≤ x ≤ 1, and 1 for x > 1. The breakpoints are at x = 0 and x = 1. At these breakpoints, we need to check for continuity and differentiability. When dealing with piecewise functions, the points where the intervals meet (the breakpoints) are the most critical to analyze for continuity and differentiability. At these points, the function's behavior can change abruptly, leading to discontinuities or non-differentiable points.

To determine the continuity of a piecewise function at a breakpoint, we need to check if the left-hand limit, the right-hand limit, and the function's value at that point are all equal. If they are, the function is continuous at that point. To determine the differentiability, we need to check if the left-hand derivative and the right-hand derivative are equal at that point. If they are, the function is differentiable at that point. In essence, piecewise functions provide a flexible way to define functions with varying behaviors across different intervals, but they also require careful analysis at the breakpoints to ensure continuity and differentiability.

When dealing with piecewise functions, assessing continuity and differentiability requires a careful examination of the function's behavior at the points where the sub-functions meet, known as breakpoints. These points are crucial because the function's definition changes, potentially leading to discontinuities or non-differentiable points. To analyze continuity at a breakpoint, say x = a, we need to ensure that the left-hand limit, the right-hand limit, and the function's value at x = a are all equal. Mathematically, this can be expressed as:

$$\lim_{x \to a^-} f(x) = \lim_{x \to a^+} f(x) = f(a)$$

If this condition holds, the function is continuous at x = a. However, continuity alone does not guarantee differentiability. For a piecewise function to be differentiable at x = a, it must first be continuous at that point. Additionally, the left-hand derivative and the right-hand derivative must also be equal. The derivative of a function at a point represents the slope of the tangent line at that point. For a function to have a well-defined tangent line, the slopes approaching from the left and right must be the same. This condition can be expressed as:

$$\lim_{x \to a^-} f'(x) = \lim_{x \to a^+} f'(x)$$

If both the continuity and differentiability conditions are met, the piecewise function is said to be smooth at x = a. However, if the function is continuous but the left-hand and right-hand derivatives are not equal, the function is not differentiable at x = a. This typically occurs at sharp corners or cusps in the graph of the function. This analysis involves evaluating the limits of the function and its derivative as x approaches the breakpoint from both sides. If the limits exist and are equal, the function is continuous and differentiable, respectively. If the function is continuous but the derivatives are not equal, it signifies a point where the function has a sharp turn, making it non-differentiable. Understanding these conditions is essential for a thorough analysis of piecewise functions.

To determine which piecewise function is continuous but not differentiable at x = -3, we must analyze the given function. The function provided is:

$$f(x)=\left\{\begin{array}{l}x^2+2 \text { if } x\ \textless \ -3 \\ x^4+3 x^3+21 x+74 \text { if } x\geq -3\end{array}\right.$$

Our goal is to ascertain if this function is continuous at x = -3 and, if so, whether it is also differentiable at the same point. To check for continuity, we need to verify if the left-hand limit, the right-hand limit, and the function's value at x = -3 are equal. First, let's find the left-hand limit:

$$\lim_{x \to -3^-} f(x) = \lim_{x \to -3^-} (x^2 + 2) = (-3)^2 + 2 = 9 + 2 = 11$$

Next, we find the right-hand limit:

$$\lim_{x \to -3^+} f(x) = \lim_{x \to -3^+} (x^4 + 3x^3 + 21x + 74) = (-3)^4 + 3(-3)^3 + 21(-3) + 74 = 81 - 81 - 63 + 74 = 11$$

Now, we evaluate the function at x = -3:

$$f(-3) = (-3)^4 + 3(-3)^3 + 21(-3) + 74 = 11$$

Since the left-hand limit, the right-hand limit, and the function's value at x = -3 are all equal to 11, the function is continuous at x = -3. Now, we need to check for differentiability. To do this, we need to find the derivatives of the sub-functions and evaluate them at x = -3. The derivative of the first sub-function, x² + 2, is:

$$f'(x) = 2x$$

The derivative of the second sub-function, x⁴ + 3x³ + 21x + 74, is:

$$f'(x) = 4x^3 + 9x^2 + 21$$

Now, we find the left-hand derivative:

$$\lim_{x \to -3^-} f'(x) = \lim_{x \to -3^-} (2x) = 2(-3) = -6$$

And the right-hand derivative:

$$\lim_{x \to -3^+} f'(x) = \lim_{x \to -3^+} (4x^3 + 9x^2 + 21) = 4(-3)^3 + 9(-3)^2 + 21 = -108 + 81 + 21 = -6$$

Since the left-hand derivative and the right-hand derivative are both equal to -6, the function is also differentiable at x = -3. Therefore, this function is continuous and differentiable at x = -3.

In this comprehensive analysis, we've explored the crucial concepts of continuity and differentiability, particularly in the context of piecewise functions. We've established that continuity is a prerequisite for differentiability, but it doesn't guarantee it. A function must be continuous at a point to be differentiable there, but the existence of a sharp corner, cusp, or vertical tangent can render a continuous function non-differentiable. Piecewise functions, with their multiple sub-functions defined over specific intervals, offer a rich landscape for examining these concepts. The breakpoints, where the function's definition changes, are the critical points for analysis. We've demonstrated how to meticulously check for continuity by ensuring the left-hand limit, right-hand limit, and function value coincide at the breakpoint. For differentiability, we've shown the necessity of equal left-hand and right-hand derivatives. The specific piecewise function presented in this article served as a practical example. Through detailed calculations, we determined that this particular function was both continuous and differentiable at x = -3. However, the broader principle remains: functions can be continuous without being differentiable, a key insight in calculus. This understanding is essential for anyone studying calculus, as it highlights the subtle yet significant differences between these two fundamental properties of functions. By mastering these concepts, one can gain a deeper appreciation for the behavior of functions and their applications in various fields.