Thread: Integrals problem

Let f be a continuous function on open interval I.

Let h>0 and a<b so that: (a-h,b+h) c I.

Now we choose d: 0 < d< h

and now for every x in(a,b) we define the following function:

G(x) = 1/2d * {int(-d --> d) f(x+t)dt}

Prove:
1. G is differentiable functions whose derivative is continuous.
2. lim(d-->0+) G(x) = f(x)





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Originally Posted by Also sprach Zarathustra

Let f be a continuous function on open interval I.

Let h>0 and a<b so that: (a-h,b+h) c I.

Now we choose d: 0 < d< h

and now for every x in(a,b) we define the following function:

G(x) = 1/2d * {int(-d --> d) f(x+t)dt}

Prove:
1. G is differentiable functions whose derivative is continuous.
2. lim(d-->0+) G(x) = f(x)

Notice that as . This is because of the continuity of f, which means that we can take for all t in the interval . So .

For 1, as .
For 2, , which you can estimate in the same way.