Let F(a, b) denote the set of real valued functions defined on the interval (a, b), C(a, b) the set of continuous real-value functions on (a, b), and D(a, b) the set of differentiable functions on (a, b). Now my book says that D(a, b) is a subset in the subspace of C but is it valid to say that C is in the subspace of D? Since not all continuous functions are differentiable and all differentiable functions are continuous. Why does my book say it the other way around or are both ways valid?
Is the set of continuous functions in the subspace of differentiable functions
linear algebra
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Best Answer
The space is continuous function is far larger than the space of differentiable functions . Think of it more carefully. If you had a function $f$ that was differentiable then $f$ is already continuous so is in $C$. But if you chose an arbitrary function $g$ that is continuous, then it is clearly in $C$ but is it in $D$? We can't say because not all continuous functions are differentiable. Thus $D$ is a subset of $C$ but not the otherway around.