# Continuity and derivability

**THEOREM**: If a function f is derivable in x = a, then f is continuous in x = a, too.

*demonstration*: we have to check that

**NOTE**: Not all continuous functions in x = a are derivable in x = a.

*derivability implies soft curves and non derivability implies peaks.*

**derivative function**as:

•We only have to study the derivability of a function at the points which the function is continuous. At these points we have to check if the lateral limits (one-sided or lateral derivatives) are equal, f’(a^{-}) = f’(a^{+}) (left derivative = right derivative).

f is derivable in R-{1}

**Exercises**:

1.- Find the derivative function of:

2.- Find the abscissa in which the slope of the tangent line to the graph of f(x) = x^{2} + 1 is 6.

3.- Study the derivability of

4.- Find the value of *a* and *b* to do the function

continuous and derivable in **R**

Solutions: 1. f'(x) = -2/x^{2}; b) f'(x) = 2x; 2. x = 3; 3. f is derivable in **R**; 4. a = 2, b = -1

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