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Prove that if a function is differentiable at $a$ then it is also continuous at $a$.

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Lesson Parent:

Problem:

Answer:

It is true that if a function is differentiable at $a$ then it is also continuous at $a$.

Solution:

Recall that

Now since

we have

Now with some trivial algebra we have

and can play the same trick again knowing that <Sign in to see all the formulas> to find

But recall from above that this equals zero so that

Now we know that

since $f(a)$ is a constant here. Now because

we know that

Finally, we know that if

then
$f(x)$ is continuous at $a$.