When is a Function Non-Differentiable?

Introduction

In calculus, differentiability refers to the ability to find a derivative, or the rate of change, at a particular point on a function. When a function is differentiable at a point, it means we can calculate its slope smoothly at that location.

However, not all functions are differentiable everywhere. Certain conditions can cause a function to be non-differentiable at specific points. In this blog, we’ll discuss four key situations where a function fails to be differentiable:

1. Discontinuity: Where the function has a break or gap
2. Corner: Where the function has a sharp turn.
3. Vertical Tangent: Where the slope becomes infinite.
4. Cusp(Bird): Where the function forms a pointed peak.

Grasping these concepts not only deepens your understanding of calculus but also prepares you for tackling complex problems in exams like AP Calculus AB, AP Calculus BC, and college-level Calculus.

Types of Discontinuity

1: Hole Discontinuity

(Both limit finite)

2: Infinite Discontinuity

and/or

3: Jump Discontinuity

(Both limit finite)

Some Examples on Non – Differentiability

1. Discontinuity

Consider the step function, also known as the Heaviside function. The function is defined as:

In this function, for values of less than 0, the function outputs 0. As soon as reaches 0 or becomes positive, the function jumps to 1. There’s no gradual change—just an abrupt jump.

Why it’s Non-Differentiable:

At , the left-hand limit of the function is 0, and the right-hand limit is 1. Since these limits don’t match, the function has a jump discontinuity at , making it non-differentiable at that point. The derivative does not exist at this point because there’s no single tangent line that can be drawn through the jump.

2. Corner: A Sharp Turn in the Graph

A corner is a point on a function’s graph where there is a sharp turn or kink. Unlike a smooth curve where the slope changes gradually, a corner has a sudden change in direction. This abrupt change means that the function does not have a single, well-defined tangent line at that point, leading to non-differentiability.

Characteristics of a Corner

At a corner, the function’s graph makes a distinct, sharp turn. Here’s what typically happens:

– Left-Hand and Right-Hand Slopes: The slopes of the function as you approach the corner from the left and from the right are different. This means that while the function may be continuous at that point, the rate of change isn’t consistent.

– No Single Tangent Line: Because the slope changes abruptly, there is no single tangent line that can be drawn at the corner. The tangent line on one side of the corner does not match the tangent line on the other side.

Example of a Corner

Consider the function defined as:

This function behaves differently depending on whether is non-positive or positive. For , the function is a parabola , which smoothly transitions. For , the function is the absolute value , which has a sharp V-shaped corner at .

Why it’s Non-Differentiable

At :

– As approaches 0 from the left, the slope of the function is 0.

– As approaches 0 from the right, the slope of the function is 1.

Since these slopes are not equal, the function does not have a well-defined derivative at . The abrupt change in direction at the corner prevents the function from having a single, consistent tangent line.

Graphical Representation

In a graph, a corner appears as a distinct point where the curve sharply changes direction. You might see a clear “V” or “L” shape in the graph, indicating the point where the function’s rate of change abruptly shifts.

Why It Matters

Corners are important in calculus because they highlight points where a function might not behave as smoothly as we might need for certain analyses. Understanding corners helps in identifying where standard calculus techniques might not apply and provides insight into the function’s behavior at critical points.

3. Vertical Tangent: An Infinite Slope

A vertical tangent occurs at a point on a function where the slope of the tangent line becomes infinitely steep. Essentially, the function’s graph has a vertical line at that point, which implies that the slope is unbounded.

Characteristics of a Vertical Tangent

At a point with a vertical tangent:

– Slope Behavior: The slope of the tangent line grows without bound as you approach the point. This means that as you get closer to the point from either side, the slope increases dramatically, approaching infinity.

– Function Behavior: The graph of the function becomes nearly vertical at that point. This vertical behavior indicates that the function changes extremely rapidly, which prevents a finite slope from being defined.

Example of a Vertical Tangent

Consider the function:

This function represents the cube root of . At , the graph of this function has a vertical tangent.

Here’s why:

– Near : The cube root function increases very quickly as approaches 0 from the left and right. As gets closer to 0, the slope of the tangent line becomes steeper and steeper, approaching infinity.

– At : The tangent line at is vertical, indicating that the derivative at this point is not finite.

Why it’s Non-Differentiable

At :

– The function’s graph is almost vertical, meaning the slope of the tangent line is infinitely large.

– Because the derivative (slope) of a function cannot be infinite, the function is non-differentiable at this point.

Graphical Representation

In a graph, a vertical tangent looks like the graph is approaching a vertical line at the point of interest. The function appears to turn sharply, making the slope grow very large, illustrating why the tangent line can’t be defined as having a finite slope.

Why It Matters

Understanding vertical tangents is crucial for analyzing functions with extreme rates of change. These points are significant in applications where rapid changes are involved, and recognizing them helps in determining where conventional differentiation might not apply.

4. Cusp: A Pointed Peak (A bird function)

A cusp is a point on a function where the graph forms a sharp, pointed peak. At a cusp, the function has a distinct, sudden change in direction, resulting in a point where the left-hand and right-hand slopes are different.

Characteristics of a Cusp

At a cusp:

– Sudden Change in Direction: The function abruptly changes direction at the cusp, forming a sharp point.
– Different Slopes: The slopes of the function as you approach the cusp from the left and right are different. This means that there is no single tangent line that fits both sides of the cusp.

Example of a Cusp

Consider the function:

This function represents the absolute value of raised to the power of . At , the function has a cusp.

Here’s why:

– Near : The function behaves differently on either side of . On both sides, the graph sharpens into a pointed peak at .

– At : The left-hand and right-hand slopes are not equal, making the tangent line undefined at this point.

Why it’s Non-Differentiable

At :

– The function has a sharp, pointed peak where the rate of change suddenly shifts.

– The left-hand derivative and right-hand derivative at this point do not match, making it impossible to define a single, continuous tangent line. Therefore, the function is non-differentiable at the cusp.

Graphical Representation

In a graph, a cusp appears as a sharp, pointed vertex where the function’s direction changes abruptly. The graph will show a clear, pointed peak at the cusp, illustrating the sharp turn and differing slopes.

Why It Matters

Recognizing cusps is important for understanding functions that have abrupt changes in direction. These points are critical in various applications, particularly in optimization problems where sharp changes in the function’s behavior can affect outcomes and analyses. Understanding cusps helps in identifying where smoothness breaks down and ensures accurate modeling of real-world phenomena.

Conclusion

Understanding differentiability is crucial for analyzing and interpreting mathematical functions. Differentiability ensures that a function’s rate of change is well-defined and consistent at a given point, allowing for the calculation of slopes and tangent lines. However, there are specific scenarios where functions fail to meet these criteria, leading to non-differentiability.

We’ve explored four primary cases of non-differentiability:

1. Discontinuity: Where a function has breaks or jumps, making it impossible to define a single slope.
2. Corner: Where a function has a sharp turn, leading to differing slopes on either side of the point.
3. Vertical Tangent: Where the function’s slope becomes infinitely steep, making it impossible to assign a finite derivative.
4. Cusp: Where a function forms a pointed peak, resulting in abrupt changes in direction and slopes.

Additionally, we highlighted that non-continuous functions cannot be differentiable at points of discontinuity, as differentiability requires continuity. However, continuity alone does not guarantee differentiability; the function must also meet smoothness criteria.

FAQs on Differentiability

Q. What Does Non-Differentiable Mean?

In calculus, when we say a function is non-differentiable at a certain point, it means that the derivative of the function does not exist at that point. Differentiability refers to the ability of a function to have a well-defined tangent line, or slope, at every point within its domain.

A function may be non-differentiable for various reasons:

• Discontinuity: The function may have a break or jump at the point, which makes it impossible to define a single slope.
• Corner: The function may have a sharp turn, where the slopes from either side do not match.
• Vertical Tangent: The function may have an infinitely steep slope, making it impossible to assign a finite derivative.
• Cusp: The function may form a pointed peak where the direction changes abruptly, leading to different slopes from either side.

In simpler terms, if a function is non-differentiable at a point, you can’t draw a single straight line that touches the function at that point and represents its rate of change.

Q. Can a Non-Continuous Function Be Differentiable?

No, a function cannot be differentiable at a point where it is not continuous. For a function to be differentiable at a point, it must first be continuous at that point. Differentiability implies continuity, but continuity alone does not guarantee differentiability.

Here’s why:

• Continuity is a Prerequisite: If a function is not continuous at a point, it means there is a break, jump, or gap in the function’s graph at that point. This discontinuity prevents the function from having a well-defined slope or tangent line at that point.
• Smoothness Required: Differentiability requires a smooth transition in the function’s behavior at the point of interest. Without continuity, this smooth transition is impossible.

If you’re unsure about the different calculus paths, you might find our blog on AP Calculus AB or BC: Which path to take? helpful.

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