Prove that if positive-definite $f$ is continuous at $0$, it is continuous on $\mathbb{R}$
Hint: The matrix $$ A=\pmatrix{ f(0) &f(-t) &f(-t-h)\\ f(t) &f(0) &f(-h)\\ f(t+h) &f(h) &f(0)} $$ is congruent to $$ B=\pmatrix{ f(0) &f(-t) &f(-t-h)-f(-t)\\ f(t) &f(0) &f(-h)-f(0)\\ f(t+h)-f(t) &f(h)-f(0) &2f(0)-f(h)-f(-h)}. $$ Now consider the $2\times2$ submatrix taken from the entries at the four corners of $B$.