Is there any shortcut method for checking continuity and differentiablity of a function without calculating the whole limits?

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You should remember that all elementary functions are continuous and differentiable functions at any point of the domain of definition.

The list of the simplest elementary functions consists of the following functions, such that: power or polynomial functions, exponential functions, logarithmic functions, trigonometric functions.

Hence, by continuity of elementary functions, you may test the continuity of a function if it represents any combination of all the elementary functions listed above.

There exists a problem if the function is defined by different equations on different parts of its domain of definition, such that:

`f(x) = {(2x - 1, x >=1),(3x - 2, x<1):}`

In this case you cannot avoid limit evaluation because the function is continuous over `(-oo,1)` or `[1,oo)` , but you need to test the continuity at the point `x = 1` .

`lim_(x->1,x<1)(3x - 2) = 3 - 2 = 1`

`lim_(x->1,x>1)(2x - 1) = 2 - 1 = 1`

`f(1) = 2 - 1 = 1`

Since `lim_(x->1,x<1)(3x - 2) = lim_(x->1,x>1)(2x - 1) = f(1) = 1` yields that the function is continuous at x = 1.

As sciencesolve points out elementary functions are continuous and differentiable on their domains.

(1) Check the domain. This is critical for rational functions (`f(x)=(p(x))/(q(x))` where p(x),q(x) are polynomials); you just need to check for zeros in the denominator.

Also be alert for functions with rational exponents.

Continuity for elementary functions is assured as long as you don't divide by zero, take an even root of a negative number, or the logarithm of a nonpositive entry.

If the function is defined piecewise using elementary functions, you normally just need to check the endpoints of the intervals of definition.

(2) If the function is continuous, it will probably be differentiable.

The problems for continuous functions occur when you have an absolute value or take even roots of even-powered entries.

For instance y=|x| is continuous everywhere, and differentiable everywhere except at x=0 as there is a "corner".

`y=sqrt(x^2)` is continuous everywhere and differentiable everywhere except at x=0 as there is a "cusp".

You must also check the endpoints of a piece-wise defined function for differentiability even if the function is continuous at those endpoints. The slope can change at those points.

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So for elementary functions you can assume continuity unless there is a division by zero, even root of a negative argument, or if the argument of a logarithm is nonpositive, and you must check the endpoints of the intervals of a piecewise function.

Also for an elementary function that is continuous you can assume differentiability unless there is an absolute value or an even root of an even powered argument. (You must still check the endpoints of a peicewise function for differentiability even if the function is continuous at those endpoints.)

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