### Differentiable and Continuous Functions

We say that $y=f\left(x\right)$  is differentiable at a point $x={x}_{0}$  if ${f}^{\prime }\left({x}_{0}\right)$  is a finite number.

A function $y=f\left(x\right)$  is called differentiable if it is differentiable at every point where it is defined.

Lets consider a differentiable function $y=f\left(x\right)$ .Then

$\frac{dy}{dx}=\underset{\Delta x\to 0}{\mathrm{lim}}\frac{\Delta y}{\Delta x}$

${f}^{\prime }\left(x\right)=\underset{\Delta x\to 0}{\mathrm{lim}}\frac{\Delta y}{\Delta x}$

${f}^{\prime }\left(x\right)\approx \frac{\Delta y}{\Delta x}$

The symbol '$\approx$' is read as almost equal to.

$\Delta y\approx {f}^{\prime }\left(x\right)\Delta x$

Now if $\Delta x\to 0$  , then clearly $\Delta y\to 0$  because ${f}^{\prime }\left(x\right)$  is a finite number. ($f\left(x\right)$  is a differentiable function, so the derivative is finite) This would not be true if the derivative ${f}^{\prime }\left(x\right)$  was not a finite number.

What this says is that if $y=f\left(x\right)$  is a differentiable function, then little changes in $x$  will generate only little change in $y$. There will not be any large arbitrary shift in the value of $y$  when $x$  changes in value just slightly. That is precisely is the definition of a continuous function.

We can conclude that a differentiable function is a continuous function.

The converse is not true. Not all continuous functions are differentiable. You may consider the absolute value function $y=f\left(x\right)=|x|$  which is continuous everywhere, yet it is not differentiable everywhere. Actually, it is differentiable everywhere except at 0.