Medium Practice — Set 2

Quick Recap — Continuity & Differentiability (JEE Application)

• Continuity of piecewise $f$ at $a$: LHL $=$ RHL $=f(a)$ — one equation per constant.
• Differentiability at $a$: LHD $=$ RHD; differentiable $\Rightarrow$ continuous (converse false — corners, cusps, vertical tangents break it).
• $|f(x)|$ is non-differentiable where $f$ changes sign; $\big[x\big]$ (greatest integer) jumps at every integer.
• For $f(x+y)=f(x)f(y)$-type functional equations, differentiate using $f'(x)=f(x)\,f'(0)$.