Continuity and Discontinuity – The Calculus of Connectedness

Hello, class! Today, we’re tackling a concept that is incredibly intuitive yet mathematically precise: continuity. We’ll answer the question: “Can I draw the graph of this function without lifting my pencil?” More importantly, we’ll learn why this property matters so much in calculus.

Part 1: The Big Idea – What is Continuity?

Think about drawing a function.

• Can you draw it in one smooth, unbroken stroke? If YES, it’s continuous.

• Do you have to lift your pencil to jump to another part? If YES, it has a discontinuity.

Formal Definition:
A function f(x) is continuous at a point $x=a$ if and only if the following three conditions:

1. f(a) exists. (The function is defined at $x=a$. There is a point to go to.)

2. $$\lim_{x\to a}f(x)$$ exists. (The function approaches a single, finite value as x gets close to $a$.)

3. $\$\$\backslash lim\_\{x\backslash to\; a\}f(x)\; =\; f(a)\$\$$. (The value the function is approaching is the same as the value it actually takes.)

If a function is continuous at every point in an interval, then the function is continuous on that entire interval.

Part 2: The Three Conditions in Action (Why All Three Matter)

Let’s see what happens when each condition fails. This is how we classify different types of discontinuities.

Discontinuity Type 1: The Hole (Point Discontinuity)

Function: $$f(x)=\frac{x^{2}-1}{x-1}​
Point of Interest: x=1

The Autopsy:

1. ✅ Does f(1) exist? No! Plugging in x=1 gives $$\frac{0}{0}$$, which is undefined. The point (1, 2) is missing—there’s a hole in the graph.

Since the first condition fails, the function is automatically discontinuous at $x=1$. Let’s check the others for curiosity:
2. ✅ $$\lim_{x\to1}f(x)$$ exists. If we simplify the function, we get $$f(x)=x+1$$for$$x\nqe1$$ the limit is 2.
3. ❌ Does the limit equal the function value? There is no function value, so this also fails.

Verdict: Discontinuous at $x=1$. This is called a Removable Discontinuity because we could “fix” the discontinuity by simply defining $f(1)=2$. The limit exists, but the function value is missing or wrong.

Discontinuity Type 2: The Jump (Jump Discontinuity)

Function: A piecewise function like:

$$g(x)= x+1 (x<2) | x-1 (x\geq2)$$​

Point of Interest: x=2

The Autopsy:

1. ✅ Does g(2) exist? Yes! g(2)=2−1=1. There is a point at (2, 1).

2. ❌ Does $$\lim_{x\to2}$$? No!

   • Left-hand limit: $$\lim_{x\to2^{-}}g(x)=3$$

   • Right-hand limit: $$\lim_{x\to2^{+}}g(x)=1$$

   • The left and right limits disagree, so the overall limit Does Not Exist (DNE).

Verdict: Discontinuous at $x=2$. This is called a Jump Discontinuity. The function “jumps” from one value to another. This is a Non-Removable Discontinuity; you can’t fix it by redefining a single point.

Discontinuity Type 3: The Vertical Asymptote (Infinite Discontinuity)

Function: $$h(x)=\frac{1}{x}$$​
Point of Interest: x=0

The Autopsy:

1. ❌ Does h(0) exist? No! Division by zero is undefined.

2. ❌ Does $$\lim_{x\to0}$$ ​h(x) exist? No! As $x$ approaches 0 from the right, the function goes to $$+\infty$$. As x approaches 0 from the left, it goes to $$-\infty$$. It does not approach a finite number.

Verdict: Discontinuous at x=0. This is called an Infinite Discontinuity. The function values become arbitrarily large (positive or negative) near the point. This is also Non-Removable.

Part 3: Why is Continuity So Important in Calculus?

Continuity isn’t just a nice geometric property. It’s the key that unlocks many concepts in calculus.

1. The Intermediate Value Theorem (IVT):
   If a function f is continuous on the closed interval $[a,b]$, then it takes on every value between $f(a)$ and $f(b)$.

   • Real-World Application: If you measure 60°F at 6 AM and 80°F at noon, you know for a fact that the temperature must have been exactly 75°F at some point in between. This seems obvious, but it only holds true if temperature is a continuous function—which it is!

2. Differentiability Implies Continuity:
   This is a huge one. If a function is differentiable at a point (meaning it has a derivative there), then it must be continuous at that point.

   • The Converse is FALSE: A function can be continuous but not differentiable. The classic example is $f(x)=|x|$ at $x=0$. It’s continuous (you can draw it without lifting your pencil), but it has a sharp corner, so it has no defined slope (derivative) there.

3. Easier Limit Evaluation:
   If you know a function is continuous at $x=a$, then finding $$\lim_{x\to a}f(x)$$ is trivial: you just plug in a! This is why we love continuous functions; they behave predictably.

Part 4: Summary and Key Takeaways

Let’s create a quick diagnostic flowchart for continuity at a point $x=a$:

1. Does $f(a)$ exist?

   • No → Discontinuous. (Hole or Asymptote)

   • Yes → Proceed.

2. Does $$\lim_{x\to a}f(x)$$ exist?

   • No → Discontinuous. (Jump)

   • Yes → Proceed.

3. Does $$\lim_{x\to a}f(x)=f(a)$$?

   • No → Discontinuous. (Removable Hole)

   • Yes → Continuous!

Classification of Discontinuities:

• Removable Discontinuity (Hole): The limit exists, but the function is not defined there (or is defined to be the wrong value).

• Jump Discontinuity: The left and right-hand limits both exist but are not equal.

• Infinite Discontinuity (Vertical Asymptote): The function values become unbounded near the point.