Suppose h(x) and g(x) differentiable at x =a Let f(x) = g(x) if x=a f(x) = h(x) is x

Question

Suppose h(x) and g(x) differentiable at x =a
Let f(x) = g(x) if x>=a
     f(x) = h(x) is x <a
Show that f(x) is diff. at x =a iff 
h(a) = g(a) and
h'(a) =g'(a)
Please, help

loser66 · Answer

$f(x) =\begin{cases}g(x) ~~if~~x\geq a\h(x)~~if~~x<a\end{cases}$

misty1212 · Answer

HI!

loser66 · Answer

Hello, help me, please

misty1212 · Answer

not sure what there is to show
you need to have $f$ be continuous, which is the same as saying 
$$g(a)=h(a)$$

loser66 · Answer

Yes, it is

misty1212 · Answer

i mean that is one part 
you can drag it out 
$$\lim_{x\to a^+}f(x)=g(a)\
\lim_{x\to a ^-}f(x)=g(a)$$ so if $g(a)=h(a)$ then the limit exists and is that number

misty1212 · Answer

i made a mistake on the second line it should be $h(a)$

loser66 · Answer

Yes, how about the opposite direction?

misty1212 · Answer

guess you are not finished though you also have to say the derivatives are the same too right?

misty1212 · Answer

oh but you are given that they are, so there is not too much to say ...

loser66 · Answer

I think derivative part is for the opposite direction, right?

loser66 · Answer

oh, I got it

loser66 · Answer

Thanks for the help

misty1212 · Answer

$$\color\magenta\heartsuit$$