# 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. 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).

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) = x2 + 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/x2; b) f'(x) = 2x; 2. x = 3; 3. f is derivable in R; 4. a = 2, b = -1