Continuity of Function — Complete Formula

Basic Definition of Continuity

Concept	Formula	Symbols Meaning	Key Notes / Tricks
Continuity at [x = a]	[\lim_{x \to a} f(x) = f(a)]	[f(x)] = function, [a] = point	Core definition ⭐

💡 Memory Hint:
Function is continuous if no break at the point

Concept	Formula	Symbols Meaning	Key Notes / Tricks
Condition 1	[f(a) \text{ exists}]	[f(a)] = value of function at [a]	Function must be defined ⭐
Condition 2	[\lim_{x \to a^-} f(x) \text{ exists}]	Left-hand limit	Approach from left
Condition 3	[\lim_{x \to a^+} f(x) \text{ exists}]	Right-hand limit	Approach from right
Final Condition	[\lim_{x \to a^-} f(x) = \lim_{x \to a^+} f(x) = f(a)]	—	All must match ⭐

💡 Memory Hint:
LHL = RHL = value → continuous

Concept	Formula / Condition	Symbols Meaning	Key Notes / Tricks
Removable	[\lim_{x \to a} f(x) \text{ exists but } \neq f(a)]	—	Hole in graph ⭐
Jump	[\lim_{x \to a^-} f(x) \neq \lim_{x \to a^+} f(x)]	—	Sudden jump ⚠️
Infinite	[\lim_{x \to a} f(x) = \pm \infty]	—	Vertical asymptote ⭐

💡 Memory Hint:

Function Type	Condition	Symbols Meaning	Key Notes
Polynomial	Continuous ∀ [x]	—	Always continuous ⭐
Rational	Continuous where denominator ≠ 0	—	Check denominator ⚠️
Trigonometric	Continuous in domain	—	Watch undefined points
Exponential	Continuous ∀ [x]	—	Smooth function ⭐
Logarithmic	Continuous for [x > 0]	—	Domain restriction ⚠️

💡 Memory Hint:
Discontinuity usually comes from denominator or domain

Concept	Formula	Symbols Meaning	Key Notes
Sum	[f + g \text{ is continuous}]	[f, g] = functions	If both continuous ⭐
Difference	[f – g \text{ is continuous}]	—	Same rule
Product	[fg \text{ is continuous}]	—	Always holds
Quotient	[\dfrac{f}{g} \text{ is continuous if } g(x) \neq 0]	—	Denominator ≠ 0 ⚠️

💡 Memory Hint:
Operations preserve continuity (except division by zero)

Concept	Formula	Symbols Meaning	Key Notes
Definition	[\|x\| = x \text{ if } x \ge 0; \quad] [\|x\| = – x \text{ if } x \le 0; \quad]

Concept	Formula	Symbols Meaning	Key Notes
GIF	[f(x) = \lfloor x \rfloor]	—	Discontinuous at integers ⭐

Concept	Formula	Symbols Meaning	Key Notes
Fractional Part	[{x} = x – \lfloor x \rfloor]	—	Jump discontinuity ⭐

Concept	Formula	Symbols Meaning	Key Notes
Condition	Match LHL, RHL and value at joining point	—	Solve for unknown constant ⭐

💡 Memory Hint:
Used to find constants (k, a, etc.) in JEE

Concept	Formula	Symbols Meaning	Key Notes
Trig Limit	[\lim_{x \to 0} \dfrac{\sin x}{x} = 1]	—	Most important ⭐
Cos Limit	[\lim_{x \to 0} \dfrac{1 – \cos x}{x} = 0]	—	Useful
Exponential	[\lim_{x \to 0} \dfrac{e^x – 1}{x} = 1]	—	Important ⭐
Log	[\lim_{x \to 0} \dfrac{\ln(1+x)}{x} = 1]	—	Standard result ⭐

💡 Memory Hint:
These limits are used to test continuity

Concept	Formula	Symbols Meaning	Key Notes
Open Interval	Continuous for all [x \in (a, b)]	—	No endpoints
Closed Interval	Continuous in [(a, b)] and at endpoints	—	Check one-sided limits ⭐

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