Introduction to Continuity and Differentiability

Welcome to one of the most fundamental chapters in Higher Level (HL) Calculus! While you may already be comfortable using rules like the power rule or the chain rule to find derivatives, this chapter is about looking "under the hood." We will explore what makes a function smooth enough to differentiate and how to find a derivative using the original definition—known as First Principles. Don't worry if these concepts seem abstract at first; we will break them down into simple, logical steps.

1. Limits, Convergence, and Divergence

Before we can talk about derivatives, we need to understand the behavior of functions as they approach certain values. This is the concept of a limit.

A limit is the value that a function \(f(x)\) approaches as \(x\) gets closer and closer to a specific number \(a\). We write this as:

\(\lim_{x \to a} f(x) = L\)

Convergence and Divergence:

  • Convergence: If the function approaches a specific, finite number as \(x\) moves toward a value (or toward infinity), we say the limit converges.
  • Divergence: If the function grows without bound (goes to \(\infty\) or \(-\infty\)) or oscillates without settling on a value, we say it diverges.
Quick Review: The Informal Idea

Think of a limit as a destination. Even if you never actually reach the point \(x = a\), the limit tells you exactly where you were heading.

2. Continuity at a Point

In simple terms, a function is continuous if you can draw its graph without lifting your pen from the paper. There are no "holes," "jumps," or "gaps."

For a function to be continuous at a specific point \(x = a\), three things must be true:

  1. The function must exist at that point (\(f(a)\) is defined).
  2. The limit must exist as you approach that point (\(\lim_{x \to a} f(x)\) exists).
  3. The limit must equal the function's value (\(\lim_{x \to a} f(x) = f(a)\)).
Common "Continuity Killers" to Watch For:
  • Vertical Asymptotes: Where the function shoots off to infinity (e.g., \(f(x) = \frac{1}{x}\) at \(x = 0\)).
  • Holes (Removable Discontinuities): Where a single point is missing from the line.
  • Jumps: Where the graph "breaks" and starts again at a different height (common in piecewise functions).

Key Takeaway: Continuity is the "baseline" requirement for many calculus rules. If a function isn't continuous, it usually can't be differentiated at that point.

3. Differentiability at a Point

Just because a function is continuous doesn't mean it is differentiable. Differentiability is a "stricter" requirement.

A function is differentiable at \(x = a\) if it has a defined gradient (slope) at that point. Geometrically, this means the graph is "smooth."

When is a function NOT differentiable?
  • Sharp Corners (Cusps): Think of the graph of \(f(x) = |x|\). At \(x = 0\), there is a sharp "V" shape. You can't decide what the slope is exactly at that corner because it changes instantly from \(-1\) to \(1\).
  • Vertical Tangents: If the graph becomes so steep that the tangent line is vertical, the gradient is undefined (\(\infty\)).
  • Discontinuities: If a function isn't continuous, it cannot be differentiable.

Analogy: Continuity is like a road with no physical gaps. Differentiability is like a road that is also smooth enough for a car to drive over without a violent jolt at a sharp corner.

4. Differentiation from First Principles

The derivative \(f'(x)\) represents the instantaneous rate of change. We find it by looking at the gradient of a secant line between two points that are incredibly close together.

The First Principles formula is:

\(f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h}\)

Step-by-Step Process:

If you are asked to differentiate a function like \(f(x) = x^2\) from first principles, follow these steps:

  1. Substitute: Plug \((x+h)\) into the function to get \(f(x+h)\).
    Example: \((x+h)^2 = x^2 + 2xh + h^2\).
  2. Subtract: Find \(f(x+h) - f(x)\).
    Example: \((x^2 + 2xh + h^2) - x^2 = 2xh + h^2\).
  3. Divide by \(h\): Simplify the fraction \(\frac{f(x+h) - f(x)}{h}\).
    Example: \(\frac{2xh + h^2}{h} = 2x + h\).
  4. Apply the Limit: Let \(h \to 0\).
    Example: \(\lim_{h \to 0} (2x + h) = 2x\).
Common Pitfall:

Don't drop the \(\lim_{h \to 0}\) notation! In IB exams, you must keep writing the limit symbol in every step until you actually substitute \(h = 0\) at the very end. If you leave it out, you may lose accuracy marks.

5. Higher Derivatives

Once you have found the derivative \(f'(x)\), you can differentiate it again! This is called a higher derivative.

  • First Derivative: \(f'(x)\) or \(\frac{dy}{dx}\) (measures the rate of change/gradient).
  • Second Derivative: \(f''(x)\) or \(\frac{d^2y}{dx^2}\) (measures the rate of change of the gradient, also known as concavity).
  • Third Derivative: \(f'''(x)\) or \(\frac{d^3y}{dx^3}\).
  • \(n^{th}\) Derivative: \(f^{(n)}(x)\) or \(\frac{d^ny}{dx^n}\).
Example:

If \(f(x) = x^4\):

\(f'(x) = 4x^3\)

\(f''(x) = 12x^2\)

\(f'''(x) = 24x\)

Did you know? In physics, if your function represents position, the first derivative is velocity, the second is acceleration, and the third is known as jerk!

Summary Checklist

  • Can you explain the difference between a jump discontinuity and a sharp corner?
  • Do you remember the 3 conditions for continuity?
  • Can you perform the algebraic expansion needed for First Principles (especially for \(x^2\), \(x^3\), or \(\frac{1}{x}\))?
  • Do you use the correct notation for higher derivatives (\(\frac{d^2y}{dx^2}\) instead of \(\frac{dy^2}{dx^2}\))?

Note: For more complex limits involving quotients or indeterminate forms, please refer to the chapter on Limits and l'Hopital's rule (HL).