Differentiability

Definition (differentiability at a point). A function $f$ is said to be differentiable

at $x=a$ if the limit

exists (finite). This is precisely the first-principles definition of the derivative carried over

from the previous chapter on limits and derivatives; here the focus shifts to the relationship

between differentiability and continuity, and to the standard rules -- chain rule, implicit

differentiation, logarithmic differentiation, parametric differentiation -- built on top of it.

Left-hand and right-hand derivatives. Exactly as with limits, a function may have a

left-hand derivative $Lf'(a)=\lim_{h\to0^-}[f(a+h)-f(a)]/h$ and a right-hand derivative

$Rf'(a)=\lim_{h\to0^+}[f(a+h)-f(a)]/h$; $f$ is differentiable at $a$ exactly when both exist and

are equal, in which case $f'(a)$ is their common value.

Theorem: differentiability implies continuity. If $f$ is differentiable at $x=a$, then $f$ is

continuous at $a$.

Proof. For $x\neq a$, write

Taking the limit as $x\to a$, and using that $f$ is differentiable at $a$ (so the first factor

tends to $f'(a)$, a finite number) together with $\lim_{x\to a}(x-a)=0$,

by the product rule for limits. Hence $\lim_{x\to a}f(x)=f(a)$, which is exactly the definition

of continuity at $a$. $\blacksquare$

The converse is FALSE. Continuity does not imply differentiability -- this is one of the

most important cautionary facts in this chapter. The standard counterexample is $f(x)=|x|$ at

$x=0$: it is continuous there (Section 1), but

Since $Lf'(0)=-1\neq1=Rf'(0)$, the two one-sided derivatives disagree, so $f$ is not

differentiable at $x=0$ -- geometrically, the graph of $|x|$ has a sharp corner at the origin,

and a corner has no single well-defined tangent line, even though the curve itself has no break

there.

Why this matters. Continuity is therefore a necessary but not sufficient condition for

differentiability: every differentiable function is automatically continuous, but a continuous

function can still fail to be differentiable, typically at a corner, a cusp, or a vertical

tangent. Every rule developed in the rest of this chapter -- the chain rule, implicit and

logarithmic differentiation, parametric derivatives -- implicitly assumes the functions involved …