<p>Prove: If the function is continuous at C and f(x) < w for every x in the domain of f, and g(x) = 1/(w-f(x)), then g is continuous at c.</p>
<p>If anyone can help me out here, I would be most appreciative! Also, I noticed the first "C" is capitalized while the second is not. Perhaps it is a typo, but if you're trying to prove it and they're supposed to be different variables or even the same variable, I'm sure it would be fine if you treated it that way. My teacher typed it up, so it may be a typo. Thanks!!</p>
<p>Does the question make sense? If anyone is confusing on the wording, I'll try to make it clearer, but that's all that was typed on the worksheet.</p>
<p>Hmm. I haven't used limits in a while, but...</p>
<p>g(x) can never be undefined unless w = f(x), which never happens because f(x) < w for all x. Therefore, g(x) is continuous at x=c. As f(x) approaches w, g(x)-> positive infinity, so I think the graph would look kind of like a e^x graph.</p>
<p>Warbles is right except that g(x) does not necessarily ever come close to w, so the graph probably wouldn't look very exponential. If w > f(x) for all values of x, w-f(x) cannot be zero, so there is no reason why g(x) would lose continuity.</p>