
Intuitively, a continuous function is one whose graph has no sudden jumps in it; the graph is all a single connected piece. Such a function can be drawn without picking the pen up off of the
paper. Formally, a function is defined to be continuous if for
any real
and any infinitesimal
,
is infinitesimal.
Example
Let the function be defined by
for
, and
for
. Then f (x ) is discontinuous, since for dx > 0, f (0 + dx ) − f (0) = 1, which isn’t infinitesimal.

If a function is discontinuous at a given point, then it is not differentiable at that point. On the other hand, the example shows that a function can be continuous without being differentiable.
In most cases, there is no need to invoke the definition explicitly in order to check whether a function is continuous. Most of the functions we work with are de- fined by putting together
simpler functions as building blocks. For example, let’s say we’re already convinced that the functions defined by and
are both continuous. Then if we encounter the function
, we can tell that it’s continuous because its definition corresponds to
. The functions
and
have been set up like a bucket brigade, so that
takes the
input, calculates the output, and then hands it off to
for the final step of the calculation. This method of combining functions is
called composition. The composition of two continuous functions is also continuous. Just watch out for division. The function
is continuous everywhere except at
, so for example
is continuous everywhere except at multiples of
, where the sine has zeroes.
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