Continuity and Differentiability
Continuity and Differentiability
This domain asks whether a function has no break at a point and whether its graph has a defined tangent there. At x=c, continuity requires lim_{x→c} f(x)=f(c); both one-sided limits must exist, agree, and equal the function value. Differentiability at c requires the derivative limit [f(c+h)-f(c)]/h as h→0 to exist.
Core rules
- Differentiability implies continuity, but continuity does not imply differentiability.
- Check piecewise functions at the joining point using left and right limits and left and right derivatives.
- Standard derivatives include d/dx(x^n)=nx^{n-1}, d/dx(sin x)=cos x, d/dx(cos x)=-sin x, and d/dx(e^x)=e^x.
- Use chain, product, quotient, implicit, and logarithmic differentiation as needed.
- Rolle’s theorem needs continuity on [a,b], differentiability on (a,b), and f(a)=f(b); then f′(c)=0 for some c in (a,b).
Worked example
Let f(x)=|x| at x=0. It is continuous because both left and right limits are 0 and f(0)=0. For h>0, [f(h)-f(0)]/h=(h-0)/h=1. For h<0, [f(h)-f(0)]/h=(-h-0)/h=-1. The one-sided derivatives differ, so f is not differentiable at 0. The graph has a corner there.
Traps
Assuming continuity proves differentiability; checking only one side of a piecewise join; forgetting to substitute the function value at the point; differentiating |x| as 1 everywhere; misapplying Rolle’s theorem when its hypotheses fail.