12:36 AM, R F wrote:

If f is differentiable at a point a, then f is continuous at a.


Proof: let e be greater than zero. since f is differentiable at a,there exists a delta 1 greater than zero such that if the distance between x and a is greater than zero and less than delta 1 then,

the distance between the difference of f(x) and f(a) divided by the difference of x and a and the derivative of f at a is less than one.therefore, if the distance between x and a is less than delta 1 and greater than 0, then the absolute value of the difference of f(x) and f(a) divided by the difference of x and a is less than 1 plus the absolute value of the derivative of f at a. set delta to equal the minimum of delta 1 and e divided by one plus the absolute value of the derivative of f at a. then if the distance between x and a is greater than 0 and less than delta, then the distance between f(x) and f(a)equals the absolute value of the difference between x and a multiplied by the difference between f(x) and f(a) divided by the difference of x and a, which is less than the distance between x and a multiplied by one plus the absolute value of the derivative of f at a, which is less than e. therefore, f is continuous at a.