Derivatives

# 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.

Example: f(x) = |x| in x = 0

As you can see,

*derivability implies soft curves and non derivability implies peaks.*We define the

**derivative function**as:We only have to study the derivability of a function at the points which the function is continuous.

Example:

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.

Solutions: 1) f'(x) = -2/x^{2}; b) f'(x) = 2x; 2) x = 3

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