Continuity and Differentiability of Functions

Continuity means the graph has no breaks; differentiability means the function has a well-defined, stable tangent line locally. Differentiability is a stronger condition than continuity—but continuity alone does not guarantee differentiability.

I first check whether the function is continuous, then verify whether the left-hand and right-hand derivatives match. Cusps and corners are especially prone to misjudgment.

In the previous section, we introduced the definition and properties of limits, learning how limits help us analyze function behavior and local characteristics. In this section, we delve deeper into continuity and differentiability—two foundational concepts in calculus that underpin the definition and geometric interpretation of derivatives.

1. Definition of Continuity

A function is continuous at a point if its graph exhibits no “jumps” or “gaps” there. More precisely, a function $f(x)$ is continuous at $x = a$ if and only if all three of the following conditions hold:

When learning continuity and differentiability, first assess whether the function values connect smoothly at the point; then determine whether a well-defined local rate of change exists. Though related, continuity and differentiability are not equivalent.

1. $f(a)$ is defined.
2. The limit $\lim_{x \to a} f(x)$ exists.
3. $\lim_{x \to a} f(x) = f(a)$.

If all three conditions are satisfied, we say $f(x)$ is continuous at $a$.

Case Study

Consider the function $f(x) = \frac{x^2 - 1}{x - 1}$. Direct substitution at $x = 1$ yields $f(1) = \frac{0}{0}$, which is undefined. So we first simplify:

$$f(x) = \frac{(x-1)(x+1)}{x-1} = x + 1 \quad (x \neq 1)$$

Now evaluate the limit as $x \to 1$:

• Compute the limit:

$$\lim_{x \to 1} f(x) = \lim_{x \to 1} (x+1) = 2.$$

Thus, although $f(1)$ is undefined, we observe:

• The limit exists and equals 2, matching the function’s value for all $x \neq 1$.

Therefore, $f(x)$ is not continuous at $x = 1$. However, we can define a new function:

$$g(x) = \begin{cases} x + 1 & \text{if } x \neq 1 \\ 2 & \text{if } x = 1 \end{cases}$$

With this definition, $g(x)$ is continuous at $x = 1$.

2. Definition of Differentiability

A function is differentiable at a point if its derivative exists there. Formally, $f(x)$ is differentiable at $x = a$ if the following limit exists:

When reading “Continuity and Differentiability in Functions and Limits”, start by reviewing the tasks, core concepts, practice prompts, and decision points illustrated in the accompanying figures—then return to the main text to fill in details. This approach helps you quickly identify which real-world scenarios this material applies to.

$$\lim_{h \to 0} \frac{f(a + h) - f(a)}{h}$$

If this limit exists, we say $f(x)$ is differentiable at $x = a$, and its derivative at that point equals the value of the limit—denoted $f'(a)$.

Case Study

Consider $f(x) = x^2$. We want to compute its derivative at $x = 1$:

1. First compute $f(1)$:

   $$f(1) = 1^2 = 1.$$

2. Then compute the difference quotient limit:

   $$f'(1) = \lim_{h \to 0} \frac{f(1+h) - f(1)}{h} = \lim_{h \to 0} \frac{(1+h)^2 - 1}{h} = \lim_{h \to 0} \frac{1 + 2h + h^2 - 1}{h} = \lim_{h \to 0} \frac{2h + h^2}{h}.$$

   Simplify further:

   $$f'(1) = \lim_{h \to 0} (2 + h) = 2.$$

Hence, $f(x) = x^2$ is differentiable at $x = 1$, and its derivative there is $2$.

3. Relationship Between Continuity and Differentiability

Crucially, differentiability implies continuity: If a function is differentiable at $a$, it must also be continuous there. The converse, however, does not hold: A function may be continuous at a point yet fail to be differentiable there. For example, $f(x) = |x|$ is continuous at $x = 0$, but not differentiable there—its left-hand and right-hand derivatives differ.

After studying “Continuity and Differentiability in Functions and Limits”, try applying it to your own scenario. Pay special attention to whether inputs, processing steps, and outputs align coherently.

To apply “Continuity and Differentiability in Functions and Limits” to your own task, begin by narrowing the scope—focus on verifying just one critical decision point.

Summary

In this section, we explored continuity and differentiability—and their essential roles in the theory of limits. Mastering these concepts strengthens your foundation in calculus and prepares you for interpreting derivatives geometrically. In the next section, we’ll examine the formal definition of the derivative and its geometric meaning—stay tuned!