If $f (x)$ and $g (x)$ both are discontinuous at any point, then show that their composition may be differentiable at that point.

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Solution

Suppose $f (x)$ and $g (x)$ are discontinuous at a point.
But, $f (g (x))$ may be continuous and differential at that point since $f (g (a + h))$ and $f (g (a - h))$ may be the same even if $g (a + h)$ and $g (a - h)$ be different.
Same is the argument for $g (f (x)) .$

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If f(x) and g(x) both are discontinuous at any point, then show that their composition may be differentiable at that point.

Suppose f(x) and g(x) are discontinuous at a point.But, f(g(x)) may be continuous and differential at that point since f(g(a+h)) and f(g(a−h)) may be the same even if g(a+h) and g(a−h) be different.Same is the argument for g(f(x)).

If $f (x)$ and $g (x)$ both are discontinuous at any point, then show that their composition may be differentiable at that point.

Suppose $f (x)$ and $g (x)$ are discontinuous at a point.
But, $f (g (x))$ may be continuous and differential at that point since $f (g (a + h))$ and $f (g (a - h))$ may be the same even if $g (a + h)$ and $g (a - h)$ be different.
Same is the argument for $g (f (x)) .$