Understanding Problems Based on Continuity and Differentiability Using Graphs

Continuity and differentiability are two fundamental concepts in calculus that describe the behavior of functions. When dealing with these concepts, graphs can be an invaluable tool for visualizing and understanding the behavior of functions. In this guide, we will explore how to use graphs to solve problems related to continuity and differentiability.

Continuity

A function $f(x)$ is said to be continuous at a point $x = a$ if the following three conditions are met:

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

In graphical terms, a function is continuous at a point if there is no break, jump, or hole in the graph at that point.

Types of Discontinuities

Type of Discontinuity	Description	Graphical Representation
Removable	A hole in the graph; the limit exists but is not equal to the function's value.	A point on the graph is missing.
Jump	The left and right limits exist but are not equal.	The graph has a sudden jump.
Infinite	The limit does not exist because it approaches infinity.	The graph has a vertical asymptote.
Oscillating	The limit does not exist due to oscillating behavior near the point.	The graph shows wild fluctuations.

Differentiability

A function $f(x)$ is differentiable at a point $x = a$ if the derivative $f'(a)$ exists. This means that there is a unique tangent to the curve at that point, and the function is smooth (no sharp corners or cusps).

Conditions for Differentiability

1. $f(x)$ must be continuous at $x = a$.
2. The left-hand derivative (LHD) and right-hand derivative (RHD) at $x = a$ must exist and be equal.

Mathematically, the LHD and RHD are defined as:

$$ \text{LHD at } x = a = \lim_{h \to 0^-} \frac{f(a + h) - f(a)}{h} $$

$$ \text{RHD at } x = a = \lim_{h \to 0^+} \frac{f(a + h) - f(a)}{h} $$

If both limits exist and are equal, then the function is differentiable at that point.

Using Graphs to Solve Problems

Graphs can help us identify points of discontinuity and non-differentiability. Let's look at some examples to illustrate these concepts.

Example 1: Continuity

Consider the function $f(x) = \frac{1}{x}$. The graph of this function has a vertical asymptote at $x = 0$.

graph LR
    A[Graph of 1/x] --> B[Vertical Asymptote at x=0]

Since the limit as $x$ approaches 0 does not exist, the function is not continuous at $x = 0$.

Example 2: Differentiability

Let's take the function $f(x) = |x|$. The graph of this function has a sharp corner at $x = 0$.

graph LR
    A[Graph of |x|] --> B[Sharp Corner at x=0]

Although the function is continuous at $x = 0$, the derivative does not exist at this point because the LHD and RHD are not equal. Therefore, the function is not differentiable at $x = 0$.

Example 3: Combining Continuity and Differentiability

Consider the piecewise function:

$$ f(x) = \begin{cases} x^2 & \text{for } x < 1 \ 2x - 1 & \text{for } x \geq 1 \end{cases} $$

The graph of this function would show a parabola for $x < 1$ and a straight line for $x \geq 1$. At $x = 1$, we need to check both continuity and differentiability.

graph LR
    A[Graph of Piecewise Function] --> B[Check Continuity and Differentiability at x=1]

To check for continuity at $x = 1$, we see if the limit from the left equals the limit from the right and the function's value:

$$ \lim_{x \to 1^-} f(x) = \lim_{x \to 1^-} x^2 = 1 $$

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

$$ f(1) = 1 $$

Since all three are equal, the function is continuous at $x = 1$.

To check for differentiability at $x = 1$, we find the derivatives from the left and right:

$$ \text{LHD at } x = 1 = \lim_{h \to 0^-} \frac{f(1 + h) - f(1)}{h} = \lim_{h \to 0^-} \frac{(1 + h)^2 - 1}{h} = 2 $$

$$ \text{RHD at } x = 1 = \lim_{h \to 0^+} \frac{f(1 + h) - f(1)}{h} = \lim_{h \to 0^+} \frac{2(1 + h) - 1 - 1}{h} = 2 $$

Since the LHD and RHD are equal, the function is differentiable at $x = 1$.

Conclusion

Understanding the concepts of continuity and differentiability using graphs can greatly simplify the process of analyzing functions. By visualizing the behavior of functions, we can quickly identify points of discontinuity and non-differentiability, which is essential for solving calculus problems. Remember that a function must be continuous to be differentiable, but continuity alone does not guarantee differentiability.