Continuity and Differentiability — Study Notes

1. Introduction

Continuity and differentiability are the two fundamental concepts of calculus that describe how smoothly a function changes. A function is continuous if its graph has no jumps, breaks, or holes, while differentiability measures the instantaneous rate of change of the function through the derivative. This chapter makes these intuitive ideas precise using limits and develops the algebraic machinery for computing derivatives.

The chapter first defines continuity at a point and on an interval, then defines the derivative as the limit of the difference quotient, and shows that differentiability implies continuity. It then develops the chain rule, derivatives of implicit and inverse functions, logarithmic differentiation, and the derivatives of all the standard functions including trigonometric, exponential, logarithmic, and parametric functions. The chapter also introduces second-order derivatives and Rolle's theorem and the Mean Value Theorem.

These tools are the engine for everything that follows: the application of derivatives in chapter 6, integration in chapter 7, and differential equations in chapter 9 all rest directly on the mastery of differentiation. Because the board examination tests both conceptual questions and routine differentiation, this chapter carries one of the highest weightages in the entire syllabus and deserves careful, systematic preparation.

2. Continuity at a Point

A function f is continuous at a point $x = a$ if the following three conditions hold: 1. f(a) is defined. 2. $\lim_{x \to a} f(x)$ exists. 3. $\lim_{x \to a} f(x) = f(a)$.

Equivalently, f is continuous at a if $\lim_{x \to a} f(x) = f(a)$. If any condition fails, f is discontinuous at a.

Continuity on an Interval

A function is continuous on an open interval (a, b) if it is continuous at every point of the interval. It is continuous on the closed interval [a, b] if it is continuous on (a, b), continuous from the right at a, and continuous from the left at b.

Algebra of Continuous Functions

If f and g are continuous at a, then so are f + g, f - g, fg, and f/g (provided g(a) $\neq$ 0). Also, polynomial functions, rational functions (where the denominator is non-zero), trigonometric, exponential, and logarithmic functions are all continuous on their domains.

3. Differentiability

The derivative of a function f at a point x is defined by the limit of the difference quotient:

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

provided this limit exists. The derivative can also be written as $\frac{dy}{dx}$, or as $f'(x) = \lim_{\Delta x \to 0} \frac{f(x + \Delta x) - f(x)}{\Delta x}$.

Differentiability Implies Continuity

If f is differentiable at a point, then f is continuous at that point. The converse is false: a function can be continuous without being differentiable. The classic example is $f(x) = |x|$ at x = 0, which is continuous but has a sharp corner there, so its left and right derivatives differ ($-1$ and $+1$).

Right and Left Derivatives

• Right-hand derivative at a: $\lim_{h \to 0^+} \frac{f(a + h) - f(a)}{h}$
• Left-hand derivative at a: $\lim_{h \to 0^-} \frac{f(a + h) - f(a)}{h}$

The function is differentiable at a exactly when both exist and are equal.

4. Standard Derivatives

$$\frac{d}{dx}(x^n) = nx^{n-1}, \quad \frac{d}{dx}(\sin x) = \cos x, \quad \frac{d}{dx}(\cos x) = -\sin x$$

$$\frac{d}{dx}(\tan x) = \sec^2 x, \quad \frac{d}{dx}(\sec x) = \sec x \tan x, \quad \frac{d}{dx}(\csc x) = -\csc x \cot x$$

$$\frac{d}{dx}(\cot x) = -\csc^2 x, \quad \frac{d}{dx}(e^x) = e^x, \quad \frac{d}{dx}(\ln x) = \frac{1}{x}$$

5. Rules of Differentiation

Sum and Constant Multiple

$$(u + v)' = u' + v', \quad (cu)' = cu'$$

Product Rule

$$(uv)' = u'v + uv'$$

Quotient Rule

$$\left(\frac{u}{v}\right)' = \frac{u'v - uv'}{v^2}, \quad v \neq 0$$

Chain Rule

If $y = f(u)$ and $u = g(x)$, then

$$\frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx}$$

The chain rule extends to compositions of any length: differentiate the outer function, then multiply by the derivative of the inner function.

6. Derivatives of Inverse Trigonometric Functions

$$\frac{d}{dx}(\sin^{-1} x) = \frac{1}{\sqrt{1 - x^2}}, \quad \frac{d}{dx}(\cos^{-1} x) = -\frac{1}{\sqrt{1 - x^2}}$$

$$\frac{d}{dx}(\tan^{-1} x) = \frac{1}{1 + x^2}, \quad \frac{d}{dx}(\cot^{-1} x) = -\frac{1}{1 + x^2}$$

$$\frac{d}{dx}(\sec^{-1} x) = \frac{1}{|x|\sqrt{x^2 - 1}}, \quad \frac{d}{dx}(\csc^{-1} x) = -\frac{1}{|x|\sqrt{x^2 - 1}}$$

7. Logarithmic Differentiation

For functions of the form $y = [f(x)]^{g(x)}$, or for products and quotients of many factors, take natural logarithms on both sides and differentiate implicitly:

$$y = u^v \implies \ln y = v \ln u \implies \frac{1}{y}\frac{dy}{dx} = v' \ln u + v\frac{u'}{u}$$

Example

If $y = x^{\sin x}$, then $\ln y = \sin x \ln x$, so

$$\frac{1}{y}\frac{dy}{dx} = \cos x \ln x + \frac{\sin x}{x}$$

$$\frac{dy}{dx} = x^{\sin x}\left(\cos x \ln x + \frac{\sin x}{x}\right)$$

8. Derivatives of Implicit Functions

When y is an implicit function of x defined by an equation such as $x^2 + y^2 = 25$, differentiate both sides term by term with respect to x, treating y as a function of x:

$$2x + 2y\frac{dy}{dx} = 0 \implies \frac{dy}{dx} = -\frac{x}{y}$$

9. Derivatives of Parametric Functions

If $x = f(t)$ and $y = g(t)$, then

$$\frac{dy}{dx} = \frac{dy/dt}{dx/dt}, \quad \text{provided } \frac{dx}{dt} \neq 0$$

10. Second-Order Derivatives

The second derivative is the derivative of the derivative:

$$\frac{d^2y}{dx^2} = \frac{d}{dx}\left(\frac{dy}{dx}\right)$$

It measures the rate of change of the slope and is central to concavity analysis and the second derivative test in chapter 6.

11. Rolle's Theorem and Mean Value Theorem

Rolle's Theorem

If f is continuous on [a, b], differentiable on (a, b), and f(a) = f(b), then there exists at least one c in (a, b) such that f'(c) = 0.

Mean Value Theorem

If f is continuous on [a, b] and differentiable on (a, b), then there exists at least one c in (a, b) such that

$$f'(c) = \frac{f(b) - f(a)}{b - a}$$

Geometrically, there is a point where the tangent is parallel to the chord joining the endpoints.

Quick Revision Tables

Table 1: Standard Derivatives

Table 2: Differentiation Rules

Table 3: Continuity vs Differentiability

Mind Map

Important Diagrams (SVG)

Diagram 1: Continuous vs Discontinuous Functions

Diagram 2: Tangent Lines and the Difference Quotient

Common Mistakes

1. Forgetting to check all three conditions of continuity at a point; for instance, the limit may exist but not equal f(a).
2. Claiming that continuity at a point implies differentiability. The converse is false; |x| is continuous at 0 but not differentiable there.
3. Applying the product rule incorrectly as (uv)' = u'v'. The correct rule is (uv)' = u'v + uv'.
4. In the quotient rule, forgetting the minus sign and the square in the denominator: (u/v)' = (u'v - uv')/v^2.
5. Using the chain rule incompletely: for a triple composition, forgetting to multiply by both inner derivatives.
6. Forgetting the factor 1/y when differentiating ln y in logarithmic differentiation; the result must multiply through by y at the end.
7. In implicit differentiation, forgetting to apply the chain rule to every term containing y, i.e., forgetting the dy/dx factor.
8. In parametric differentiation, computing dy/dx = (dx/dt)/(dy/dt); the correct quotient is (dy/dt)/(dx/dt).
9. Writing d/dx(ln x) for all bases; d/dx(log_a x) = 1/(x ln a), not 1/x.
10. Forgetting the absolute value in d/dx(sec^(-1)x) = 1/(|x| sqrt(x^2 - 1)); the sign changes for negative x.

Exam Tips

1. To prove continuity at a point, compute the left-hand limit, right-hand limit, and f(a); show they are equal.
2. To test differentiability at a point, compute the left and right derivatives; they must be equal. Use this to find constants in piecewise functions.
3. When differentiating products with three factors, write u, v, w and use (uvw)' = u'vw + uv'w + uvw'.
4. For y = u^v, always use logarithmic differentiation; never attempt the power rule directly.
5. For parametric functions, first compute dy/dt and dx/dt separately, then divide; simplify only at the end.
6. Verify differentiability implies continuity, and use the contrapositive: if f is discontinuous at a, then f is not differentiable there, so no derivative needs computing.
7. When applying Rolle's theorem or the mean value theorem, first verify the hypotheses are satisfied, then set up and solve the equation for c.
8. Practise differentiating implicitly with fractional and negative powers, and always solve for dy/dx explicitly at the end.

Conclusion

Continuity and differentiability are the conceptual pillars on which the entire calculus course rests. Continuity formalises the smoothness of a function, while differentiability quantifies its rate of change as the slope of the tangent line. The algebraic rules for differentiation, together with the special techniques of implicit, logarithmic, and parametric differentiation, equip the student with a complete toolkit for computing derivatives of any combination of standard functions. The second derivative adds information about curvature, and Rolle's theorem and the mean value theorem provide the theoretical guarantees that power the optimisation and curve analysis of the next chapter. Mastery of this chapter transforms the subsequent chapters on application of derivatives, integrals, and differential equations from a struggle into a natural application of skills already acquired.