Continuity is a property that captures the intuitive idea of a function having no "breaks" in its graph. For JEE, the formal definition and its implications are essential.

A function f is continuous at a point x = a if and only if three conditions hold simultaneously: (1) f(a) is defined, (2) lim(x->a) f(x) exists (meaning LHL = RHL), and (3) lim(x->a) f(x) = f(a). If any one condition fails, f is discontinuous at a.

A function is continuous on an open interval (a, b) if it is continuous at every point in (a, b). It is continuous on a closed interval [a, b] if it is continuous on (a, b) and additionally lim(x->a+) f(x) = f(a) and lim(x->b-) f(x) = f(b).

The algebra of continuous functions states: if f and g are continuous at x = a, then f + g, f - g, f * g, and f/g (provided g(a) != 0) are all continuous at x = a. The composition of continuous functions is continuous.

Types of discontinuity:

1. Removable: The limit exists but the function value is missing or wrong. This can be "fixed."
2. Jump (first kind): Both one-sided limits exist but are unequal.
3. Infinite (second kind): At least one one-sided limit is infinite.
4. Oscillatory: The limit doesn't exist due to oscillation (like sin$\frac{1}{x}$ at 0).

Important continuous functions: all polynomials (on R), rational functions (where defined), sin x, cos x, $e^x$ (on R), ln x (on (0, infinity)), |x| (on R).

Important discontinuous functions: [x] at every integer, {x} at every integer, 1/x at x = 0, tan x at x = (2n+1)*pi/2, sec x at x = (2n+1)*pi/2.

The Intermediate Value Theorem states: if f is continuous on [a, b] and k is between f(a) and f(b), there exists c in (a, b) with f(c) = k. The most common application is proving root existence: if f(a) and f(b) have opposite signs and f is continuous, there's a root between a and b.