Continuity and Differentiability — Lesson
1) Hook — A Fun Real-Life Example
Imagine you are driving on a highway from Delhi to Agra. You want to know if your speed changes smoothly or suddenly at any point. If your speedometer needle moves without any sudden jumps, your speed is continuous. But if you suddenly hit the brakes, causing an abrupt drop, that’s a point where speed is not continuous. Similarly, differentiability tells us if the speed changes smoothly without sharp turns or corners. This is exactly what Continuity and Differentiability help us understand in mathematics!
2) Core Concepts — Continuity and Differentiability Explained
- \( f(a) \) is defined.
- \( \lim_{x \to a} f(x) \) exists.
- \( \lim_{x \to a} f(x) = f(a) \).
Example: Consider \( f(x) = \frac{x^2 - 9}{x - 3} \) at \( x = 3 \).
| Step | Calculation | Result |
|---|---|---|
| Is \( f(3) \) defined? | \( \frac{3^2 - 9}{3 - 3} = \frac{0}{0} \) (undefined) | No |
| Simplify \( f(x) \) | \( \frac{(x-3)(x+3)}{x-3} = x+3 \), \( x \neq 3 \) | \( f(x) = x+3 \) for \( x \neq 3 \) |
| Limit at 3 | \( \lim_{x \to 3} f(x) = \lim_{x \to 3} (x+3) = 6 \) | 6 |
Conclusion: \( f(x) \) is not continuous at 3 because \( f(3) \) is not defined. But if we define \( f(3) = 6 \), then \( f \) becomes continuous at 3.
- If \( f \) is differentiable at \( a \), then \( f \) is continuous at \( a \).
- But continuity at \( a \) does not guarantee differentiability at \( a \).
Example: \( f(x) = |x| \) at \( x = 0 \)
- Left-hand derivative: \( \lim_{h \to 0^-} \frac{|0+h| - |0|}{h} = \lim_{h \to 0^-} \frac{-h}{h} = -1 \)
- Right-hand derivative: \( \lim_{h \to 0^+} \frac{|0+h| - |0|}{h} = \lim_{h \to 0^+} \frac{h}{h} = 1 \)
Since left and right derivatives are not equal, \( f \) is not differentiable at 0, even though it is continuous there.
3) Key Formulas / Rules
Continuity at \( x = a \):
\( \lim_{x \to a} f(x) = f(a) \)
Differentiability implies continuity:
If \( f \) is differentiable at \( a \), then \( f \) is continuous at \( a \).
Derivative definition:
\( f'(a) = \lim_{h \to 0} \frac{f(a+h) - f(a)}{h} \)
Derivative of standard functions:
- \( \frac{d}{dx} (x^n) = n x^{n-1} \)
- \( \frac{d}{dx} (\sin x) = \cos x \)
- \( \frac{d}{dx} (\cos x) = -\sin x \)
- \( \frac{d}{dx} (e^x) = e^x \)
- \( \frac{d}{dx} (\ln x) = \frac{1}{x} \), \( x > 0 \)
4) Did You Know?
In India, the concept of continuity and smooth change was intuitively used in ancient mathematics and astronomy. For example, Aryabhata’s astronomical calculations required smooth functions to predict planetary positions, which implicitly assumed continuity and differentiability long before formal calculus was developed!
5) Exam Tips — Common Mistakes and Board Patterns
- Always check three conditions for continuity: existence of \( f(a) \), existence of limit, and equality of both.
- Do not confuse continuity with differentiability: A function can be continuous but not differentiable (e.g., \( |x| \) at 0).
- When given piecewise functions, check continuity and differentiability at the junction points carefully.
- Use the definition of derivative for tricky points: Board often asks to prove differentiability using the limit definition.
- Previous Year Question Pattern:
- Prove/disprove continuity or differentiability at a point (3-4 marks).
- Find \( a \) and \( b \) for piecewise functions to be continuous/differentiable (4 marks).
- Application-based problems involving tangent or rate of change (3 marks).
- Practice tip: Sketch graphs to visualize continuity and sharp corners — it helps in understanding differentiability.
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