Continuity & DifferentiabilityJEE Main

Holes, jumps, corners and blow-ups — interactive Mathematics simulation for IIT-JEE.

Concept

Continuity at a point needs THREE things: the left limit, the right limit, and the value — all equal. Fail modes have names: limits agree but value missing (removable), limits disagree (jump), limits blow up (infinite). Differentiability is stricter still: |x| is continuous everywhere yet its corner at 0 has no unique tangent.

Key formula

\lim_{x\to a^-}f = \lim_{x\to a^+}f = f(a) \iff \text{continuous at } a; \quad \text{differentiable} \Rightarrow \text{continuous (not conversely)}

Derivation

sin(x)/x → 1 as x → 0 but f(0) is undefined — patch it (define f(0)=1) and it's continuous: removable.

|x|/x jumps from −1 to +1 — no patch can fix a jump. Differentiability: the limit of [f(a+h) − f(a)]/h must exist; at |x|'s corner the left slope (−1) ≠ right slope (+1).

Scenarios to explore

Continuity & Differentiability — Holes, jumps and kinks — LHL, RHL and f(a) face off.

Real-world applications

Piecewise-defined functions: find constants making them continuous/differentiable.
IVT (roots exist) and EVT (max/min exist) both REQUIRE continuity.
Physical laws break at discontinuities — shocks, impacts.

JEE exam tips

For piecewise f with parameter k: equate LHL = RHL = f(a) — one equation per continuity demand, slopes for differentiability.
x·sin(1/x) → 0 (squeeze theorem) — continuous at 0 once defined.
Differentiable ⇒ continuous is a one-way street exams love to reverse.

Common mistakes

Continuous ⇒ differentiable (FALSE — |x|).
Checking only the limit and forgetting f(a) itself.
Assuming both one-sided limits exist because the formula 'looks fine'.

Exam traps to avoid

⌊x⌋ is right-continuous at integers, not continuous.
Weierstrass functions are continuous EVERYWHERE, differentiable NOWHERE — cited in theory MCQs.