# continuity

#### janae77

Let f: [0,1]$$\displaystyle \rightarrow$$R be defined as:

f(x) = 1/$$\displaystyle \sqrt{x}$$ - $$\displaystyle \sqrt{((1+x)/x)}$$ if x is not equal to 0.

I need to prove that f is continuous on [0,1]. What I want to do is pick a point in [0,1] and show that is is continuous at that point. So i can pick p to be .5. Then f(x)-f(.5) is less than epsilon but the algebra here i am having a hard time. Can someone please help me. Thank you .

#### tonio

Let f: [0,1]$$\displaystyle \rightarrow$$R be defined as:

f(x) = 1/$$\displaystyle \sqrt{x}$$ - $$\displaystyle \sqrt{((1+x)/x)}$$ if x is not equal to 0.

I need to prove that f is continuous on [0,1]. What I want to do is pick a point in [0,1] and show that is is continuous at that point. So i can pick p to be .5. Then f(x)-f(.5) is less than epsilon but the algebra here i am having a hard time. Can someone please help me. Thank you .

$$\displaystyle \frac{1}{\sqrt{x}}-\sqrt{\frac{1+x}{x}}=\frac{1}{\sqrt{x}}-\frac{\sqrt{1+x}}{\sqrt{x}}$$ $$\displaystyle =\frac{1-\sqrt{1+x}}{\sqrt{x}}$$

Now, this is a quotient of functions so it is continuous at any point where both the numerator and the denominator are continuous and the latter isn't zero, so

the only problem is at $$\displaystyle x=0$$ . Since the function isn't defined at this point, I suppose you meant to show that this point is a removable discontinuity and for that

you need to show the limit of the function exists and is finite there:

$$\displaystyle \lim_{x\to 0}\frac{1-\sqrt{1+x}}{\sqrt{x}}$$ , and it's easy to show this limit is zero (with L'Hospital, for example).

Tonio