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Let alpha in R. prove that a function...

Let `alpha in R`. prove that a function `f: R-R` is differentiable at `alpha` if and only if there is a function `g:R-R` which is continuous at `alpha` and satisfies `f (x) -f(alpha) = g (x) (x-alpha), AA x in R.`

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Since, g(x) is continuous at `x = rArr underset( x to alpha) lim g(x) = g (alpha) and f(x)-f (alpha) = g (x) (x-alpha), AA x in R` [given]
`rArr" "(f(x)-f(alpha))/((x-alpha)) = g (x) `
`rArr" " underset( x to alpha) lim (f(x)-f(alpha))/(x - alpha) = underset( x to alpha) lim g (x) `
`rArr" " f'(alpha) = underset( x to alpha) lim g(x) rArr f'(alpha) = g(alpha)`
`rArr ` f(x) is differentiable at ` x = alpha`.
Conversely, suppose f is differentiable at `alpha`, then
`underset( x to alpha) lim (f (x) - f(alpha))/(x-alpha) ` exists finitely.
Let `g(x) = {{:((f(x)-f(alpha))/(x-alpha)", " x ne alpha),(f'(alpha)", " x = alpha):}`
Clearly, ` underset( x to alpha) lim g (x) = f' (alpha) `
Hence, f(x) is differentiable at `x = alpha`, iff g (x) is continuous at ` x = alpha`.
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