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

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

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

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

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

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

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

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

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

What this says is that if $y=f(x)$ 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 $y=f(x)=\left|x\right|$ which is continuous everywhere, yet it is not differentiable everywhere. Actually, it is differentiable everywhere except at 0.