Continuity & Differentiability
(KVPY – Mathematics – 2024 / “Historic”)
These notes are designed for quick revision and for building intuition. They contain concise definitions, key theorems, a worked example, and a summary that you can read in a few minutes.
1. Continuity
| Concept | Simple Explanation |
|---|---|
| Continuity at a point | A function f is continuous at a if tiny changes in x near a produce tiny changes in f(x); formally lim_{x→a} f(x) = f(a). |
| Continuity on an interval | f is continuous on I if it is continuous at every point of I. |
| Types of discontinuities | 1. Removable – limit exists but ≠ f(a) (fix by redefining f(a)).2. Jump – left & right limits exist but differ. 3. Infinite – limit is ±∞. |
| Equivalences | f is continuous at a ⇔ lim_{x→a} f(x) = f(a) ⇔ `∀ε>0 ∃δ>0: |
Key Theorems
| Theorem | Statement |
|---|---|
| Intermediate Value Theorem (IVT) | If f is continuous on [a,b] and y lies between f(a) and f(b), then ∃ c∈(a,b) with f(c)=y. |
| Extreme Value Theorem | Continuous f on closed bounded interval [a,b] attains a maximum and a minimum. |
| Uniform Continuity | f is uniformly continuous on S if for every ε>0 there exists a single δ that works for all x,y∈S. Every continuous function on a compact set is uniformly continuous. |
2. Differentiability
| Concept | Simple Explanation |
|---|---|
| Derivative at a point | f'(a) = lim_{h→0} (f(a+h)-f(a))/h (if the limit exists). It gives the slope of the tangent line. |
| Differentiability implies continuity | If f is differentiable at a, then f is continuous at a. (But converse is false.) |
| Endpoints | A function can be differentiable at an interior point only. Right‑hand and left‑hand derivatives may exist at endpoints (one‑sided). |
| Non‑differentiable points | Points where the derivative fails to exist (e.g., sharp corners, vertical tangents). |
Basic Rules
| Rule | Formula |
|---|---|
| Constant | (c)' = 0 |
| Power | (x^n)' = n x^{n-1} |
| Sum / Difference | (f±g)' = f' ± g' |
| Product | (fg)' = f'g + fg' |
| Quotient | (f/g)' = (f'g - fg')/g^2 |
| Chain | (f∘g)' = (f'∘g)·g' |
Mean Value Theorem (MVT)
If
fis continuous on[a,b]and differentiable on(a,b), ∃c∈(a,b)such that
f'(c) = (f(b)-f(a))/(b-a).
Rolle’s Theorem is a special case (when f(a)=f(b)).
3. Darboux Property
Derivative of a function satisfies the Intermediate Value Property (IVP) even if it is not continuous.
Proof sketch: Apply MVT tofon[x, y]to showf'takes every intermediate value betweenf'(x)andf'(y).
4. Common Pitfalls
| Mistake | Why It Happens | Fix |
|---|---|---|
| Assuming continuity → differentiability | Sharp corners or cusps may exist. | Check if the left & right limits of the difference quotient agree. |
| Ignoring one‑sided limits at endpoints | Some problems ask for derivative at a but a is boundary. |
Use right‑hand or left‑hand derivative definition. |
| Misusing MVT when function not differentiable inside | MVT requires interior differentiability. | Verify differentiability before applying. |
5. Worked Example
Problem
Show that
[
f(x)=
\begin{cases}
x^2\sin\frac{1}{x}, & x\neq 0\[4pt]
0, & x=0
\end{cases}
]
is differentiable at x=0, and compute f'(0).
Solution
- Check continuity at 0
[ \lim_{x\to0}x^2\sin\frac{1}{x}=0 \quad\text{(because }|x^2\sin(1/x)|\le x^2\to0). ]