Continuity and Differentiability
Continuity and Differentiability tests limits, continuity at a point, differentiability, standard derivatives, chain rule, implicit differentiation, and parametric forms. The syllabus also includes Rolle’s and Lagrange’s mean value theorems, so hypotheses matter.
Core rules
f is continuous at c if lim x→c f(x)=f(c), equivalently LHL=RHL=f(c). Differentiability at c requires the left and right derivatives to exist and agree. Differentiability implies continuity, but continuity does not imply differentiability. Use product, quotient and chain rules correctly. For implicit relations, differentiate both sides with respect to x and solve for dy/dx. For parametric x=x(t), y=y(t), dy/dx=(dy/dt)/(dx/dt), and d²y/dx²=(d/dt(dy/dx))/(dx/dt). Rolle’s theorem needs continuity on [a,b], differentiability on (a,b), and f(a)=f(b); then f′(c)=0 for some c.
Worked example
Test f(x)=|x| at x=0. It is continuous because lim x→0 |x|=0=f(0). But the left derivative is −1 and the right derivative is +1. Since they differ, f is not differentiable at 0. This shows continuity alone is insufficient.
Traps
Do not assume f′(c)=0 at every point where a graph has a corner; the derivative may not exist. When differentiating sin(3x²), include the inner derivative 6x. In parametric second derivatives, do not divide by dx/dt twice by mistake. For mean value theorems, state the interval and verify all hypotheses before concluding.