174 Chapter 15

A better definition, but one that in my view remains problematic, is another phrase one

commonly hears in an elementary calculus class:

A function f is continuous at c if the closer and closer x gets to c, the closer and

closer f (x) gets to f (c).

This phrase is an improvement, since it makes reference to what I find to be the central idea

of continuity, namely, the idea that one can obtain increasingly good approximations to the

value of a continuous function by applying it to increasingly good approximations to the

input. But still, the definition is vague. What exactly does it mean? I claim, furthermore,

that a careful reading of this definition will reveal it to be incorrect. Namely, consider the

fact that as I walk north from Times Square in New York to Central Park, getting closer

and closer to the park, I am also getting closer and closer (ever so slightly) to the North

Pole. The problem is that, even though I am getting closer and closer to the North Pole,

nevertheless I am not getting close to the North Pole, since Central Park is thousands of

miles away from the North Pole, and no part of it is close to the North Pole. The suggested

definition does not sufficiently distinguish between the idea of getting closer and closer to

a quantity and the idea of getting close to it. How close do we want to get? How close

suffices? The definition does not say.

Consider the elevation function of a backpacker hiking atop a gently sloped plateau,

slowly descending toward the edge, where a dangerous cliff abruptly drops. As she de-

scends the slope toward the cliff’s edge, she is getting closer and closer to the edge, and

her elevation is getting closer and closer to the elevation of the valley floor (since she is

descending, even if only slightly), but the elevation function is not continuous, since there

is an abrupt vertical drop at the cliff’s edge, a jump discontinuity. Similarly, for the dis-

continuous red function on the previous page, as x approaches the location c of the jump

discontinuity from either side, then for x sufficiently close to c, the value of f (x) does get

closer and closer to f (c), even though when approaching from the left, they do not get

close to f (c).

For these kinds of reasons, a more correct definition should not refer to “closer and

closer” but, rather, should make a precise and explicit mention of exactly how close we

want f (x)tobeto f (c) and, furthermore, how close it will suffice for x to be to c in order

to ensure this. And this is precisely what the usual epsilon-delta definition of continuity

achieves:

Definition 131. A real-valued function f defined on an interval or all of the real numbers

is continuous at a point c if, for every positive >0, there is δ>0 such that every x within

δ of c has f (x) within  of f (c). A function is continuous if it is continuous at every point

in its domain.

One may express the continuity of f at

c succinctly in symbols as

∀>0 ∃δ>0 ∀x |x − c| <δ=⇒|f (x) − f (c)| <.