Prove Continuity of g: A Continuity Problem Homework Statement

[e(ho0n3]

Homework Statement
Let f be a function with the property that every point of discontinuity is a removable discontinuity. This means that $$\lim_{y\to x} f(y)$$ exists for all x, but f may be discontinuous at some (even infinitely many) numbers x. Define $$g(x) = \lim_{y\to x} f(y)$$. Prove that g is continuous.

The attempt at a solution
So I have to prove that for all a,

$$\lim_{x \to a} g(x) = \lim_{x \to a} \lim_{y\to x} f(y) = g(a) = \lim_{y\to a} f(y)$$

In other words, for every e > 0, there is a d > 0 such that

$$\left| \lim_{y\to x} f(y) - \lim_{y\to a} f(y) \right| < e$$

for all x satisfying |x - a| < d. I have no clue how to find d. Any tips.

 

[HallsofIvy]

I think you are going to need to reduce the two limits in your last inequality to their definitions.

 

[e(ho0n3]

OK. So let d' be such that |f(y) - g(x)| < e for all y with |y - x| < d' and let d'' be such that |f(y) - g(a)| < e for all y with |y - a| < d''.

The problem now is that I can't fiddle around with |f(y) - g(x)| < e and |f(y) - g(a)| < e because there are different intervals where these inequalities are true.