lim(x→x0)f(x)=f(x0)

The continuity of a function in a certain interval means

Any x0 that belongs to the interval has the above formula.

There is another important conclusion: the elementary function is continuous in its meaningful domain.

From the image, differentiable function is a smooth curve, that is, there is no sharp point. For example, the absolute value of y=x is a sharp point at x=0, so it is not differentiable. Moreover, because differentiability must be continuous, discontinuous points (discontinuous points) must be nondifferentiable.

According to the definition, f'(x0)=lim△x→0.

[f(x0+△x)-f(x0)]/△x

We must find the function f(x)

The necessary and sufficient condition for the derivative at x=x0 is that both the left and right derivatives exist and are equal at x=x0, that is, f'(x0-0)= f'(x0+0).