In this explainer, we will learn how to determine whether a function is differentiable and identify the relation between a functionโs differentiability and its continuity.
Differentiation is hugely important, and being able to determine whether a given function is differentiable is a skill of great importance. The derivative of a function measures the rate of change of the function value with respect to its input and is often described as the โinstantaneous rate of change.โ
When we learn about derivatives, we learn two important facts: firstly, that the derivative is the slope of the tangent to the graph of a function at any given point and, secondly, that the derivative is defined by a limit and, therefore, only exists if the given limit exists. Using these two important ideas about derivatives, we can determine whether certain derivatives exist. We begin by recapping the definition of the derivative in terms of a limit.
Definition: The Derivative of a Function
The derivative of a function at a point is defined as
An alternative but equivalent definition of the derivative is
We say that a function is differentiable at if these limits exist. If only the left or right limit exists, then we say that the function is differentiable at from the left or right.
We note that for a function , the derivative can also be written as which reads as โthe derivative of with respect to โ or โ by .โ
A differentiable function is a function whose derivative exists at each point in its domain. In other words, if is a point in the domain, then is differentiable at if and only if the derivative exists and the graph of has a nonvertical tangent line at the point .
We can also have a notion of differentiability when a function is defined on an interval. If a function is differentiable on some open set , this means that the function is differentiable for all points . For a closed interval , the function cannot be differentiable at as the limit would only exist when the function is defined on an open interval about ; nevertheless, a function is differentiable on when it is differentiable on and differentiable from the right at and from the left at .
In the definition above, we mentioned that the derivative is defined as a limit, if the limit exists, which indicates that it is possible that the limit does not exist. In such cases, we say that the function is not differentiable at this point. In this explainer, we will explore the relationship between the continuity of a function and the differentiability and consider different ways in which a function can fail to be differentiable.
Since the derivative at a point represents the slope of the tangent to the curve at that point, this tells us that if we are unable to define a tangent to a curve, the derivative will not exist. The first case we will consider is the case where the function is discontinuous. If the function has a jump discontinuity, we will not be able to define a tangent to the curve at that point. We would therefore expect that the derivative will not be defined at such a point.
In the first example, letโs consider the differentiability of a piecewise function with a jump discontinuity.
Example 1: Differentiability of a Function with a Jump Discontinuity
Suppose
What can be said of the differentiability of at ?
Answer
In this example, we want to examine the differentiability of a piecewise function at a particular point.
The given piecewise function is composed of two smooth (differentiable) functions. Generally, in situations like this, the derivative of the function is composed of the derivative of the functions defining each part. However, we also need to consider whether the functions agree at the points where the functions are joined together. If we apply this approach, we can differentiate each part of this function using the power rule as follows:
We can then consider the points where these two functions join and find that the derivative on either side of is . At this point, we might naively conclude that the function is differentiable. However, this is not the correct answer. To see this, we will use the definition of the derivative, and demonstrate that this limit does not exist at . Since the function is defined differently on each side of the point , we will consider the left and right limits. Beginning with the left limit, we have
Using the definition of the function , we have and can rewrite the limit as
Since , we can cancel this common factor from the numerator and denominator to get
We can now consider the right limit,
Using the definition of the function we have been given, we have
We can split this fraction and rewrite this as
In this case, since , the limit does not exist. The reason the limit does not exist is the fact that the function is actually discontinuous at this point as we can see from its graph.
Since the limit does not exist, the derivative is not defined. Hence, we can say that the function is not differentiable at .
The last example showed that the function was not differentiable at the point of discontinuity. This is actually a general result, that at the points where a function is discontinuous it is not differentiable. Hence, in the last example, the most efficient solution is to first confirm that the function is not continuous and therefore not differentiable.
Now, letโs consider an example where we examine the differentiability of a function from its graph that has a corner at a point.
Example 2: Assessing the Differentiability of a Function from Its Graph
The figure shows the graph of . What can be said of the differentiability of at ?
Answer
In this example, we want to assess the differentiability of a function at a particular point from its graph.
The graph of the function has a corner at the point where . This means that the slope of the tangent to the left of is not equal to the slope of the tangent to the right of . Hence, the derivative will have a jump discontinuity and will not be defined at this point since its right and left limits will not agree. Hence, The function is not differentiable at because the functionโs rate of change is different on both sides of that point.
There are many examples of functions whose graphs have corners. Two common types of functions that can have corners are functions defined piecewise or functions defined in terms of the absolute value. In the next example, we will consider another case where the derivative is not defined.
Example 3: The Existence of a Derivative at a Cusp
The figure shows the graph of . At which points is the derivative of the function not defined?
Answer
In this example, we will examine the existence of a derivative of a function that contains cusps.
The graph shows a function with two cusps, one at and one at . At these cusps, the tangent to the curve is vertical. When the tangent is vertical, its slope is infinite, which will also imply that the limit does not exist. Hence, the derivative of this function is not defined at the points and .
The last example showed us that the derivative is not defined at a cusp of a real-valued function. More generally, if the tangent to a curve is vertical, the derivative is not defined. The next example will highlight one such function.
Example 4: Domain of the Derivative
Consider the function .
- What is the domain of ?
- Find an expression for the derivative of .
- What is the domain of the derivative ?
Answer
In this example, we will consider the domain of the derivative, or where the derivative is well defined, for a cube root function.
Part 1
The real cube root of any real number is well defined. Therefore, the domain of is all the real numbers .
Part 2
We can find an expression for the derivative of by using the power rule which states that
Hence,
Part 3
To find the domain of the derivative, we need to consider the points for which is not defined. The only point where this is not defined is when the denominator is zero. This occurs when . Therefore, the domain of is all real which we can write as .
The previous example shows that the derivative of a continuous function might fail to exist at certain points in the domain. In particular, if the tangent line of a function is vertical, the derivative will not exist at this point.
Now, letโs consider an example where a function is not differentiable at a point as a result of infinitesimal oscillations.
Example 5: Oscillatory Functions and Derivatives
Is the function differentiable at ?
Answer
In this example, we want to assess the derivative of a piecewise oscillatory function at a particular point.
To assess the differentiability of this function at , we will consider the existence of the following limit:
Using the definition of the function , we have
Since , we can cancel it from the numerator and denominator to get
This is an example of a limit that does not exist due to the oscillatory behavior of the function. Therefore, the limit does not exist. Hence, is not differentiable at . A graph of demonstrates that the function displays a high level of oscillatory behavior near the origin, which is the reason why the derivative does not exist.
We have looked at many examples of how functions can fail to be differentiable. In many cases, these were continuous functions. Hence, we have seen that we can have continuous functions that are not differentiable. In fact, it is possible to have functions that are continuous everywhere but nowhere differentiable. The first known example of such a function was the Weierstrass function. Although functions such as the Weierstrass function seem unusual, it can be shown mathematically that the vast majority of continuous functions are actually nowhere differentiable! For example, a function with a bend, cusp, or vertical tangent may be continuous but fail to be differentiable at the location of the anomaly.
Even though it is possible (and, rather intuitively, very common) to have continuous functions which are not differentiable, the converse is not true; all differentiable functions are continuous as we will demonstrate below.
Let be a function that is differentiable at a point . Then, by definition,
We will show that is continuous by demonstrating that
We begin by considering the limit
By multiplying and dividing by , we have
Using the rules of finite limits, we can rewrite this as
Using the definition of the derivative above, we have
Furthermore, we know that . Therefore,
Once again, we can use the rules of finite limits to get
Therefore,
Since is independent of , the limit on the right-hand side is and we have
Hence, we have shown that a function is continuous at all points where it is differentiable; differentiability implies continuity.
While we have seen that the converse does not hold (i.e., a continuous function does not need to be differentiable), we can still use this result to deduce that if a function is discontinuous at a point, then it is not differentiable at that point.
In our final few examples, we will apply what we have learned about the existence of derivatives and the connection between differentiability and continuity.
In the next example, we will use the fact of differentiability to deduce a particular limit.
Example 6: Functions and Derivatives
Consider a function with and . What is ?
Answer
In this example, we will determine the value of the given limit of a function by using the connection between differentiability and continuity.
We have been told that ; therefore, we know that is differentiable at . Since differentiability implies continuity, we know that
Hence, .
In the final two examples, we will consider functions defined piecewise. When dealing with such functions, it is important to check for continuity, and then we can differentiate each part separately and consider the points where parts meet.
In the next example, we will assess the differentiability of a piecewise function at a particular point.
Example 7: Assessing the Differentiability of a Function
Suppose
What can be said of the differentiability of at ?
Answer
In this example, we will assess the differentiability of the given piecewise function at a particular point.
We will begin by ensuring that the function is continuous at . From the definition, we can see that ; furthermore, we can see that
Therefore, the function is continuous at . We can now apply the power rule to differentiate each part of the function as follows:
We now need to consider the left and right limits to ensure they agree. From the definition of , we can see
Therefore, we can conclude that the limit exists and . Hence, the function is differentiable at .
In the last example, we will determine the missing values of the parameters in a continuous piecewise function and assess its differentiability at a given point.
Example 8: Assessing the Differentiability of a Function
Find the values of and and discuss the differentiability of the function at given that is continuous and
Answer
In this example, we will assess the differentiability of the given piecewise function at a particular point.
Since is continuous, it is continuous at ; therefore,
Therefore,
Hence, . Furthermore,
Therefore, . Hence,
We can now consider the derivative on each side of using the power rule as follows:
We now need to consider the left and right limits at . From the definition of , we can see
Therefore, the left and right limits do not agree and the function is not differentiable at .
We will finish this explainer by recapping some of the important concepts.
Key Points
- The derivative of a function at is defined as An alternative but equivalent definition of the derivative is
- A function is not differentiable when this limit does not exist. This can happen
in a number of different ways including the following.
- If a function is differentiable, then it is continuous. The contrapositive of this statement (which is logically equivalent and consequently equally true) is that a function is not differentiable at the points where it is discontinuous.
- There are many continuous functions that are not differentiable.