If f has a derivative at x=a, then f is continuous at x=a. True or False?

Question

Hello,

I am still trying to understand the concept of derivatives conjoined with limits. I believe the answer is true (I'm more than likely wrong), but does anyone have an explanation for it on the basis whether it's true or false.

Thanks.

Also, an explanation for the converse of this statement.

Answer 1

yes, it is true
read here:
http://planetmath.org/?op=getobj&from=objects&id=6154

for the converse, it is not true, counterxample
f(x)=|x|, the absolute value of x
it is continuos in 0 but not differentiable

Answer 2

if x is differentiable at point a then it must have an instantaneous rate of change at that point , in other words how can there be a slope if there is no point of which to get the slope of thus x must be continous at a TRUE

let me further explain

if there was no derivative at point a then you could have an equation like this y= x/(x-a) , where x= a would result in x/0 = nonexistant and thus not diferentiable

however if y is diferentiable at a the it is continous at a

Answer 3

Yes the answer is true; differentiability implies continuity (the converse isn't always true however).