Intermediate Value Theorem (IVT):
Let f be continuous on a closed interval [a,b]. Pick a y-value M with f(a)f(b) or f(b)f(a). Then there is some x-value c with a < c < b and f(c) = M. We will use "IVT" interchangeably with Intermediate Value Theorem.
Here's what's going on, in pictures. Start with a continuous function f on a closed interval [a,b]:
Mark on the y-axis where f(a) and f(b) are:
Pick any value of M strictly in between f(a) and f(b):
Draw a horizontal dashed line at height y = M. The IVT guarantees that this dashed line will hit the graph of f. In other words, the IVT guarantees the existence of some value c strictly in between a and b where the function value is M:
We've said a continuous function is one we can draw without lifting our pen from the paper. The IVT states this more precisely. If a continuous function starts at f(a) and ends at at f(b), then as x travels from a to b the function must hit every y value in between f(a) and f(b):
If a function on [a,b] skips a value, that function must be discontinuous: