Cue Column:

• Three conditions for continuity?
• How to check piecewise functions?
• What about composition?

Notes Column: A function f is continuous at x = a if and only if all three conditions hold simultaneously:

1. f(a) is defined (the function has a value at a)
2. lim(x->a) f(x) exists (LHL = RHL)
3. lim(x->a) f(x) = f(a) (the limit equals the function value)

For piecewise functions, check continuity at each junction point:

• Compute LHL using the left piece
• Compute RHL using the right piece
• Check if LHL = RHL = f(a)

Properties: If f and g are continuous at a, then f+g, f-g, f*g are continuous at a. f/g is continuous at a provided g(a) != 0. If f is continuous at a and g is continuous at f(a), then g(f(x)) is continuous at a.

Every polynomial is continuous everywhere. Rational functions are continuous wherever the denominator is non-zero. sin(x), cos(x), $e^x$ are continuous for all real x. log(x) is continuous for x > 0.

Summary: Always check all three conditions. For JEE, piecewise function problems asking "find k for continuity" are the most common — equate LHL = RHL = f(junction point).