6.2 · Continuity and the tangent-line preview of calculus

Continuity connects the limit and the actual value

A function is continuous at $\displaystyle a$ when $\displaystyle f(a)$ is defined, its finite two-sided limit exists, and the limit equals $\displaystyle f(a)$. Repairing a removable hole means assigning the existing limiting value at that input.

A jump or an infinite discontinuity cannot be repaired by changing only one point value. At an endpoint of a closed domain, continuity uses the one-sided limit from within the domain.

Secant slopes approach a tangent slope

The average rate of change from $\displaystyle a$ to $\displaystyle a+h$ is $\displaystyle [f(a+h)-f(a)]/h$ for nonzero $\displaystyle h$. If these rates approach a finite common value from both directions, their limit is the derivative at $\displaystyle a$.

For $\displaystyle f(x)=x^2$, expanding and canceling gives the difference quotient $\displaystyle 2a+h$, whose limit is $\displaystyle 2a$. This is a preview of differentiation, not a reason to substitute $\displaystyle h=0$ before simplifying.

$$f'(a)=\lim_{h\to0}\frac{f(a+h)-f(a)}{h}$$

A continuous corner can lack a tangent slope

The function $\displaystyle |x|$ is continuous at $\displaystyle 0$, but its left slopes are $\displaystyle -1$ and right slopes are $\displaystyle 1$. There is no finite two-sided derivative there.

Differentiability implies continuity; the converse fails. Use the graph to identify corners and breaks, then use one-sided slopes or limits to justify the conclusion.