#math/calculus
For the points $(x_{0},f(x_{0}))$ and $(x_{0}+\Delta x,f(x_{0}+\Delta x))$, the **difference quotient** is the slope of their secant line. A function is **differentiable at $x_{0}$** when the following limit exists:
$
f'(x_{0})
=\lim_{\Delta x\to 0}\frac{f(x_{0}+\Delta x)-f(x_{0})}{\Delta x}.
$
Equivalently, with $x=x_{0}+\Delta x$,
$
f'(x_{0})=\lim_{x\to x_{0}}\frac{f(x)-f(x_{0})}{x-x_{0}}.
$
Differentiability at $x_{0}$ implies continuity there because
$
f(x_{0}+\Delta x)-f(x_{0})
=\Delta x\,\frac{f(x_{0}+\Delta x)-f(x_{0})}{\Delta x}\longrightarrow 0.
$
The converse is false: $f(x)=|x|$ is continuous at $0$ but is not differentiable there because its left- and right-hand slopes differ.