Continuity & Differentiability

Continuity

Definition

• A function is continuous if its graph has no break or jump.
• A function which is not continuous is called a discontinuous function.

Continuity at a Point

• A function f(x) is continuous at x = a if:
• lim_x→a f(x) = f(a)
• Equivalently:
• LHL at x = a = RHL at x = a = f(a)

Left & Right Continuity

• Left continuous at x = a if lim_x→a⁻ f(x) = f(a)
• Right continuous at x = a if lim_x→a⁺ f(x) = f(a)
• Continuous at x = a ⇔ left continuous and right continuous

Continuity in an Interval

• Continuous in (a, b) if continuous at every point in (a, b)
• Continuous in [a, b] if:
• Continuous in (a, b)
• Right continuous at x = a
• Left continuous at x = b

Examples of Continuous Functions

• f(x) = x (Identity function)
• f(x) = c (Constant function)
• Polynomial functions
• Trigonometric functions: sinx, cosx
• Exponential functions: aˣ, eˣ, e⁻ˣ
• Logarithmic function: log x
• Hyperbolic functions: sinhx, coshx, tanhx
• Absolute value functions: |x|, x|x|, x + |x|, x − |x|

Discontinuous Functions

• A function is discontinuous if it is discontinuous at at least one point in its domain.

Common Discontinuous Functions

• [x] → discontinuous at all integers
• x − [x] → discontinuous at all integers
• 1/x → discontinuous at x = 0
• tan x, sec x → x = π/2 ± nπ
• cot x, cosec x → x = nπ
• sin(1/x), cos(1/x) → x = 0
• e^1/x → x = 0
• coth x, cosech x → x = 0

Properties of Continuous Functions

• Sum, difference, product of continuous functions are continuous
• Quotient is continuous if denominator ≠ 0
• Scalar multiple of a continuous function is continuous
• Composite of continuous functions is continuous

Types of Discontinuity

• Removable discontinuity:
• Limit exists but ≠ f(a)
• Non-removable discontinuity:
• LHL ≠ RHL or limit does not exist

Differentiability

Differentiability at a Point

• f(x) is differentiable at x = c if:
• lim_x→c [f(x) − f(c)] / (x − c) exists finitely
• This limit is the derivative of f at x = c
• Denoted by f′(c), Df(c) or (df/dx)_x=c

Left Hand & Right Hand Derivatives

• Left hand derivative (LHD):
• Lf′(c) = lim_h→0⁻ [f(c+h) − f(c)] / h
• Right hand derivative (RHD):
• Rf′(c) = lim_h→0⁺ [f(c+h) − f(c)] / h
• f(x) is differentiable at x = c ⇔ Lf′(c) = Rf′(c)

Differentiability in an Interval

• f(x) is differentiable in (a, b) if it is differentiable at every point of (a, b)

Important JEE Main Notes

• Differentiability ⇒ Continuity
• Continuity ⇏ Differentiability (e.g., |x| at x = 0)
• Check LHL, RHL first for continuity
• Then check LHD, RHD for differentiability
• Piecewise functions are commonly tested