What is Continuity?
Informally, a function is continuous at a point if you can draw its graph at that point without lifting your pen. There should be no holes, jumps, or breaks in the graph.
No Hole
The function must be defined at the point (f(a) exists).
Limit Exists
The left-hand and right-hand limits must be equal.
They Match
The limit value must equal the function value.
Formal Definition
A function f(x) is said to be continuous at x = a if all three conditions hold simultaneously:
f(a) is defined
The function must have a value at x = a. (No holes!)
lim f(x) exists as x → a
LHL = RHL → Left-hand limit must equal right-hand limit.
lim f(x) = f(a)
The limit value must equal the actual function value.
x→a⁻ x→a⁺
Left-Hand & Right-Hand Limits
Left-Hand Limit (LHL): The value f(x) approaches as x comes from the left side of a.
Right-Hand Limit (RHL): The value f(x) approaches as x comes from the right side of a.
x→a⁻
RHL = lim f(x) = lim f(a + h) [h → 0⁺]
x→a⁺
💡 Key Point: If LHL ≠ RHL, the limit does not exist at that point → function is discontinuous there.
f(2) = 2(2) + 3 = 7 ✓ (defined)
LHL: lim(x→2⁻) 2x+3 = 2(2)+3 = 7
RHL: lim(x→2⁺) 2x+3 = 2(2)+3 = 7
LHL = RHL = f(2) = 7 → Continuous at x = 2
Types of Discontinuity
| Type | What Happens | Example |
|---|---|---|
| Removable | Limit exists but ≠ f(a), or f(a) not defined | f(x) = (x²−1)/(x−1) at x=1 |
| Jump | LHL ≠ RHL (limit doesn't exist) | Greatest Integer Function ⌊x⌋ |
| Infinite | Function → ±∞ at that point | f(x) = 1/x at x = 0 |
f(1) = 1 + 1 = 2
LHL = lim(x→1⁻) (x+1) = 1+1 = 2
RHL = lim(x→1⁺) (3−x) = 3−1 = 2
LHL = RHL = f(1) = 2 → Continuous at x = 1
f(1) = 1+2 = 3
LHL = lim(x→1⁻) (x+2) = 3
RHL = lim(x→1⁺) (x−1) = 0
LHL ≠ RHL → Discontinuous at x = 1 (Jump discontinuity)
Continuity on an Interval
A function f is continuous on an open interval (a, b) if it is continuous at every point in that interval.
A function f is continuous on a closed interval [a, b] if:
Continuous on (a, b)
Continuous at every interior point.
Right-continuous at x = a
lim(x→a⁺) f(x) = f(a)
Left-continuous at x = b
lim(x→b⁻) f(x) = f(b)
💡 Always Continuous Functions: Polynomials, Rational functions (where denominator ≠ 0), Trigonometric functions, Exponential & Logarithmic functions — all are continuous on their domains.
Algebra of Continuous Functions
If f and g are continuous at x = a, then the following are also continuous at x = a:
• f ± g is continuous
• f · g is continuous
• f / g is continuous (if g(a) ≠ 0)
• k · f is continuous (k is a constant)
• f(g(x)) — composite — is continuous if g is continuous at a and f is continuous at g(a)
⚠️ Finding k for continuity: A common exam problem gives a piecewise function and asks for the value of k that makes it continuous. Set LHL = RHL = f(a) and solve for k.
For continuity: LHL = RHL
LHL = k(5)+1 = 5k+1
RHL = 3(5)−5 = 10
5k + 1 = 10 → 5k = 9 → k = 9/5