Question

Can someone explain when functions are continuous, differentiable and defined?

Answer 1

continuous when \[\lim_{x \rightarrow c^-}f(x) = lim_{x \rightarrow c^+}f(x)\]\[\lim_{x \rightarrow c}f(x) = f(c)\]f(c) is defined, that is the denominator does not equal 0

Answer 2

um, not quite sure how to say when something's differentiable...

Answer 3

CONTINUOUS - no gaps in valid x values, no vertical asymptotes, no point (hole) discontinuities, nowhere the function is undefined.
DIFFERENTIABLE - essentially this means SMOOTH, no pointy bits.
Polynomials, trig functions, summations of those two combined like taylor series, Maclauren series etc.... tha'ts why those series are used. Log & exponentials are smooth too.

A function will not be differentiable at a point where the DERIVATIVE is discontinuous, because that means the gradient of the function approaching the point from one side has a different value as you approach the point from the other side - so you get a pointy bit!

DEFINED - you can actually evaluate the function. A function is NOT DEFINED where you have 0 in the bottom of a fraction, where you have a negative under a square root, where you have a negative as the argument of a log etc....

Answer 4

The deriveative is the gradient - so pointy bits don't have a defined gradient (slope of the tangent) - which side to you take the tangent from, as you approach the point? Get it?

Answer 5

so a function is always continuous where it is differentiable, but not always differentiable where it is continuous?

Answer 6

and if it's not differentiable, then it's not defined?

Answer 7

Sorry - I'm in a different tim-zone to you.
A function can be continuous, but not differentiable (has a pointy bit),
but it is not differentiable at a point discontinuity, or at the end points of a non-point gap discontinuity. If a function is differentiable over an interval, then it MUST automatically be continuous over that same interval.

Differentiable means it IS continuous.
Continuous does not necessarily means it's differentiable.

Answer 8

A function can be said continous if
L.H.L=R.H.L. i.e. left hand limit =right hand limit
Mathematical expr..
\[\lim_{h \rightarrow 0}f(x-h)=\lim_{h \rightarrow0}f(x+h)\]
where h = the number value around which you need to show the continuity of a function
For a function to be differentiable
\[\lim_{h \rightarrow0}(f(x-h)-f(x))/(-h)=\lim_{h \rightarrow 0}(f(x+h)-f(x))/h\]
Where h = the number value around which you need to show the diferentiability. of a function.hope you understand,if not ask