Let y=f(x) be a unary function. If the left and right derivatives of y exist and are equal at x=x0, then y is said to be derivable at x=x[0]. If a function is differentiable at x0, it must be a continuous function at x0.

Let f(x) be defined in x0 and its vicinity, then if the limit of [f(x0+a)-f(x0)]/a exists, it is said that f(x) is derivable in x0.

If F(x) is derivable at any point m on the interval (a, b), it is said to be derivable at (a, b). The differentiability of a function on the definition domain requires certain conditions: the left and right derivatives of the function at this point exist and are equal, and the existence of the derivative cannot be proved. Only when the left and right derivatives exist and are equal and continuous at this point can the point be proved to be differentiable. Derivable function must be continuous; Continuous functions are not necessarily derivable, and discontinuous functions must not be derivable.

In calculus, derivative is a measure of the tangent slope of a function. Functions with derivatives are called derivable functions. Derivability of function is an important concept in calculus and mathematical analysis. If the function is differentiable, then the derivative exists and is uniquely determined. The first is to judge the continuity of the function, whether the limit exists and whether the function is discontinuous. If the conditions are not met, it is impossible to be derivable.

Then, whether the left and right limits of the derivative are equal or not can be judged, and the conclusion whether it is derivable or not can be drawn. Finally, if the function is smooth, then the function is derivable. It should be noted that a function can only be called a derivable function if all conditions are met, otherwise it is not derivable.

Matters needing attention

1, for some unconventional functions or at some special points, differentiability needs to be judged in a deeper way.

2. Differentiability and continuity of functions are different concepts. A continuous function is not necessarily derivable, and a derivable function is not necessarily continuous.

3. When judging whether a function is differentiable, we need to pay attention to the special points or discontinuous points in the domain of the function, which may affect the judgment of the differentiability of the function.